The missionaries addressed several different topics in their preaching, but if there was a pattern or sequence, I have not yet found it. ?Authority? was a relatively frequent subject of preaching, discussion, and disputation. 
The missionaries recorded giving sermons about ?authority? by itself or in combinations: organization (5 times), baptism (1), Holy Ghost (2), restoration (2), and punishment (1).  Authority also appeared as a topic in a Sunday School lesson, a letter from the Mission President, and a sermon by the Church President. 
When the missionaries wrote that they spoke on ?authority,? they seem to have meant what in present-day LDS usage is usually specified as ?prophetic? or, more often, ?priesthood? authority: ?I talked with them until 10 o?clock, mostly upon Authority, showing them very plainly that Joseph Smith was called directly from God.?  Anecdotally and tentatively, I?d say that the missionaries emphasized institutional authority over priesthood authority. 
Conflicts sometimes centered on the question of authority. Elder Jones argued with RLDS preachers about succession: ?They did not believe that Joseph Smith had ever received the revelation on plural marriage and if he did he was a false prophet. We had it on authority. We did not have any of our books to prove to them of the authority of Brigham Young.?  Jones also ?wound?up? a Methodist ?on Baptism and the Holy Ghost. After supper he brought up the subject again. We then downed him on Authority and other Principles.?  Jones and companion ?got into an argument on the authority being equal of the first presidency and the apostles.? 
Sometimes missionaries had ?authority duels? with local individuals over the right to preach. ??Bro. Harrington, a church man, came in, and in a loud rough manner, he said, ?By the authority of the Baptist Church, I forbid you preaching here.? We thanked him and said we were acting upon the authority of the trustees.? 
The missionaries also wrote about other forms of authority. Sister Cluff and companion ?had the spirit of the Lord in rich abundance and our words, being given with all the authority and power we could muster, were listened to with great interest.? 
The ?Southwestern States Mission? series uses the diaries of
six seven eight (as of 2012 Jun 17) missionaries who served in eastern Texas or, in the case of the Sister missionaries, in Kansas and Missouri, around 1900 to illustrate aspects of Mormon material culture, lived religion, and social History. The missionaries are Mission President Duffin, Elders Brooks, Clark, Folkman, Forsha, and Jones, and Sisters Carling and Cluff. The series is inspired by Ardis Parshall?s serial posting of the missionary diary of Willard Larson Jones at Keepapitchinin. Previous installment here.
 I have not tabulated all the topics of discussion. ?Authority? shows up as a topic nineteen times and my impression is that, among the small fraction of meetings when a topic is specified, that?s pretty often. The word ?authority? appears in 36 entries: Brooks (2), Clark (9), Duffin (6), Folkman (3), Forsha (0), Jones (13), Cluff (3).
 ?Authority? shows up as a topic of a sermon nineteen times. In eleven, it?s the only topic, e.g., ?Elder Shipp took up authority and spoke for 25 minutes? (Folkman, 1901 Jan 21 Mon); ?Elder Clark done the Preaching Spoke on Divine Authority? (Clark, 1900 Dec 12 Wed). The formula is consistently ?took up authority? or ?spoke on authority? with occasional ?divine authority? and one ?spoke on the authority message? (Clark, 1900 Mar 16 Fri). Note that I counted sermons and not meetings, so ?I spoke for 55 minutes upon the principles of Baptism, and the Holy Ghost. ? After I sat down, Elder Craner talked 25 minutes on Authority? (Jones, 1902 Jan 08 Wed) gets counted as one sermon with the sole topic of ?authority.? Later analyses will consider whole meetings.
For the balance, ?authority? shared the lectern: ?authority and the neccesity of the same organization now as in the primitive church? (Folkman, 1901 Jul 14 Sun); ?the organization and divine authority? (Jones, 1900 Jul 26 Thu); ?Authority and organization? (Brooks, 1900 Mar 17 Sat); ?Authority Organization? (Brooks, 1900 Apr 02 Mon); ?Holy Ghost, Authority, and Church Organization? (Jones, 1900 Feb 04 Sun); ?baptism, Holy Ghost, and authority? (Jones, 1900 Feb 11 Sun); ?Authority, Restoration and Punishment? (Jones, 1902 Feb 01 Sat); ?authority and restoration of the Gosple? (Folkman, 1901 Feb 10 Sun).
 ?In Sunday School we had a chapter on authority, and it was thoroughly discussed by members of the whole class.? (Cluff, 1905 Jun 25 Sun); ?This letter should have been entered in July. Kansas City, Mo. July 20, 1904. Dear Brethren, – ? We should make clear to the people that this a new dispensation- one in which every principle, ordinance, blessing, power and authority of the gospel of Jesus Christ essential to the salvation of man has been restored to the earth, and that Joseph Smith was chosen and appointed of God to open up this great work upon the earth. There is nothing to be gained by a weak and vascillating course. ?? (Duffin, 1904 Aug 13 Sat); ??Prest. Joseph F. Smith: The thing of the greatest importance to the Latter-day Saints is this : “Is Mormonism true”? In order for it to be true, there must be authority, and that from God, divine authority. It is true that God has spoken in our day, has spoken to man and has declared His Son, introduced Him to the boy Joseph, and informed Joseph that he was a chosen instrument to open this dispensation. If this were not true we could be no better than the world. But this being true the Latter-day Saints hold the keys of salvation. We should get this fixed in our minds. It is true that God has restored the authority of the priesthood, and we should know it. ? Furthermore there are ordinances necessary. We came to this earth to learn to do the will of God and to gain experience. Man came into the world alone, but God ordained that man nor woman should remain singly and alone. It is essential that they be united by the authority of God, otherwise they will enter into the other world singly and alone and will be as the angels. Baptism is essential but it will avail nothing unless there is repentance, obedience and authority. With these there is a birth into the kingdom of God, without them there is no birth. ?? (Duffin, 1903 May 3 Sun).
 Jones, 1901 Jan 09 Wed. ?By request I give him the history of the Book of Mormon He said we gave good evidence concerning it and had opened his eyes on the Succession of the authority of Setting up of the Kingdom of God in the last days? (Clark, 1901 Mar 19 Tue). ?I spoke first, took up the authority and the neccesity of the same organization now as in the primitive church.? (Folkman, 1901 Jul 14 Sun); ?Among the many things spoken by [Church] Prest. [Lorenzo] Snow were the following: ??I know that Joseph Smith was a Prophet of God; ? The Lord has shown me most clearly and completely that he was a Prophet of God, and that he held the Holy Priesthood and the authority to baptize people for the remission of their sins and to lay hands upon them for the reception of the Holy Ghost, that they might receive a knowledge of the things for themselves.?? (Duffin, 1900 Oct 8 Mon).
In present-day usage, ?authority? is usually discussed under the headings of Prophets, Apostasy, Restoration, and/or Priesthood. For example, Preach My Gospel, ?The Great Apostasy,? p 35; ?The Restoration of the Gospel of Jesus Christ through Joseph Smith,? p 36; ?Priesthood and Auxiliaries,? p 83-84. (Salt Lake City: The Church of Jesus Christ of Latter-day Saints, 2004). In the diaries, the missionaries recorded teaching about ?Apostasy? and ?Restoration,? though never in combination with ?Authority?; I find no specific references to teaching about ?Priesthood.?
 That is, when not speaking about Apostles, they seem to have emphasized authority based on position within a (divine) organization rather than based on power from God directly to an individual. Duffin records giving officers of a Sunday School ?authority to hold meetings with the saints? and visiting Apostle George A Smith encouraged Elders to ?put [them]selves in the hands of the presiding authority.? Discussions of the organizations of Sunday Schools, Relief Societies, and Conference Presidencies seem to have followed a similar pattern. Obviously (I hope), I?m making an impressionistic claim, subject to more rigorous analysis later. ?After the meeting I organized the students as follows: Elmer E. Hinkley president, Julius B. Nelson and George F. Ashly, counselors, James R. Shupe, Secy., and gave them instructions and authority to hold meetings with the saints. All of these students have been on missions.? (Duffin, 1903 Dec 6 Sun); ?Priesthood Meeting: ? Apostle Geo. A. Smith: ?. Elders should have the desire to go where they are appointed to go. The Elders will be preserved if they will go where they are sent and use good judgement. Put yourselves in the hands of the presiding authority and you will be blessed where you go.? (Duffin, 1904 Sep 18 Sun).
 ?At night we came to one of the members of the Reorganized Church of Jesus Christ of L.D.S. We got the privilege of staying with them over night. After supper his father came over. They were both preachers. They did not believe that Joseph Smith had ever received the revelation on plural marriage and if he did he was a false prophet. We had it on authority. We did not have any of our books to prove to them of the authority of Brigham Young. We talked until 12 o?clock when we retired.? (Jones, 1901 Jan 11 Fri).
 ?Were taken in by Bro. J.M. Calhoun, a Methodist. [¶] Before supper we wound him up on Baptism and the Holy Ghost. After supper he brought up the subject again. We then downed him on Authority and other Principles. After he became convinced that he could do nothing with us, he says, ?Now, my boys, I want to give you a little advice. You want to be very careful, my boys, how you are teaching the people.?? (Jones, 1902 Jan 14 Tue).
 ?I rested fine, was late when we got up. Had a talk upon the gospel. At 9 a.m. we commenced our labor among the people. Stopped in the woods & had prayers. While talking we got into an argument on the authority being equal of the first presidency and the apostles. Elder Rogers said that J.M. Grant and another one had never been ordained to an apostle yet he held the same authority? (Jones, 1901 Jan 12 Sat).
 ?Duels? is my word; I haven?t noticed it in the diaries. ?Most of the crowd followed us. In a few moments, Bro. Harrington, a church man, came in, and in a loud rough manner, he said, ?By the authority of the Baptist Church, I forbid you preaching here.? We thanked him and said we were acting upon the authority of the trustees. He then began abusing the Prophet Joseph Smith?? (Jones, 1902 Jan 05 Sun); ?He being one of the trustees, he gave us the privilege of preaching in the school house, and he being the only one at home, we acted on his authority.? (Jones, 1902 Jan 04 Sat); ??at 4 o’clock we go on the East Side of the Court House and Set our grips down and was going to Sing and open meeting on the Street when the geneter [janitor?] and another man come to us and ordered us off the Court house property he was So excited he could hardly talk. one of them said you had better get out of town I ask him by what authority he had to Say that he said Shut up and don’t you talk to me, and begun to abuse us and call us murders in killing Some Emigrants that was going through Utah and John Dee Lee was killed for it and he give me a push and said go. So we go to the hotel and Stay until after Supper. ?? (Clark, 1901 Aug 28 Wed); Sometimes the Elders asserted Christian authority rather than that of secular leaders: ?just before we leave the man’s House we Stayed with his Brother came in and said what are you men doing in the country I said Preaching the Gospel he ask by what authority do You do it I told him By the authority of Jesus Christ.? (Clark, 1901 Oct 12 Sat).
 Cluff, 1905 Feb 27 Mon. President Duffin ?Spoke with Great Power and authority? (Clark, 1901 Jun 02 Sun). Also: ?It is a sad commentary on the justice of men in authority in the government that we all love, when men like Apostles Teasdale and Taylor must go into exile, while high positions in government are occupied by men, corrupt as ever lived. But we shall be patient and await our Father’s time to set right these wrongs.? (Duffin, 1905 May 14 Sun).