Baby blessings were a significant ecclesiastic duty for male missionaries. The mission kept formal records of only three ordinances: baptism, confirmation, and the naming/blessing of children. From 1897 to 1903 missionaries in Texas named and blessed approximately one child for every three baptisms they performed.  All six of the male missionaries in this study recorded performing the ordinance.
When Elder Jones was set apart as a missionary (1899), he was instructed that “if there were any babies brought to us, we were told to bless them.”  In a 1902 meeting with the First Presidency of the Church, Mission President Duffin broached the topic of “Recording names of children blessed not belonging to members of the church” and was told that “their record should be kept in separate book.” 
Missionary itinerancy made planning blessings difficult. In the first few years of Church efforts in Texas (re-starting in 1897 after the Civil War withdrawal), only missionaries held the Melchizedek Priesthood necessary to formally bless a baby, and members might go months without seeing missionaries. Further, without standing congregations to create and perpetuate expectations, I imagine that for a significant number of members, watching their own children’s blessings was the first time they had never seen—and possibly ever heard of—a Mormon baby blessing.
At any rate, if there was a pattern or process to the scheduling of baby blessings—or any indication of how missionaries broached the topic—I have not detected it in the diaries. The blessings were important enough to write down but often appear in the diaries in what seems to me an odd mix of spontaneity and run-of-the-mill-ness:
After Breakfast I milked the Cow, helped Bro Gross grind his ax, and went with him to the woods to get a piece of timber for an ax handle. After dinner we held a sacrament meeting with the Family and I blessed Sister Lena Jones’s Baby. After which we ate a water Melon and talked a few Minutes and started for Bro Carters. (Forsha, 1900 Jul 18 Wed)
We slept quite late. Didn’t eat any breakfast. Went down to Jessie’s place. Got Jessie to cut my hair. We were going to preach that evening at Jessie’s. We blessed their baby. Gave it the name of Edgar Henderson Cude. (Brooks, 1901 Jan 27 Sun)
Weather a little cool. Got ready for preaching and waited for Bro. and Sister Gross. Blessed their baby then went to the school house. Held meeting with 20 in attendance. (Folkman, 1901 Mar 31 Sun) 
Note in the Folkman entry above that the blessing was not at the meeting. Other baby blessings were performed in conjunction with baptisms or conferences or, at the least, seem to have been pre-arranged:
Stayed at Bro. Tom Odom’s all day. At to [two] o’clock we had baptizing where we held meeting for a short time. Spoke on baptism. Then I baptized nine people. After we went to Bro. Tom Odom’s where we confirmed them and blessed 10 children. We spent the rest of the day and night in talking and singing. (Folkman, 1900 Apr 10 Tue) 
As priesthood was conferred on more and more local members and as local branches were organized, the naming and blessing of children faded from the list of primary missionary duties.
The “Southwestern States Mission” series (homepage) examines mission life in (mostly) Texas around 1900.
 A first-draft count from the “Central States Mission Record of members 1886-1908” indicates that from the start of 1897 to the end of 1903 Mormon missionaries in Texas performed something like 791 baptisms (followed by confirmation); in the same period they named and blessed approximately 275 children, giving a ratio of 2.88. For comparison: on my mission, though I contributed to dozens of baptisms, I did not perform the ordinance of naming and blessing a child once.
 Jones, 1899 Nov 01 Wed. “…instructed to lay aside all bad habits that we had been used to. We were told not to baptize any woman without the consent of her husband and not to baptize children under eight years without the consent of their parents, but if there were any babies brought to us, we were told to bless them.”
 Duffin, 1902 Apr 24 Thu.
 Two more examples: “After Breakfast. we held a Sacrament meeting with Bro Bass and Family and Blessed his children that were not old enough to be baptized after which we started to visit a friend Mr Geo Griffen who lived about three miles south” (Forsha, 1900 Jun 04 Mon); “The next morning we went to the place Elder Dana called his Texas Ma’s. She was a lady he had baptized by the name of Kirkendal. We ate dinner with them. After dinner we went and seen some more of Elder Dana’s Mormons. Blessed two of their children. Elder Dana one and I the other. We then went back to Kirkendalls’ Elder Dana baptized one of their sons. I confirmed him. That made 5 Elder Dana had baptized in that settlement. We stopped overnight with Kirkendalls. Ate three meals. Weather cold.” (Brooks, 1900 Feb 15 Thu).
 Three more examples: “After eating breakfast we walked back to Paron’s as he asked us to bless his children before we left. We blessed four. Remained with him until after dinner.” (Brooks, 1900 Sep 20 Thu); “In addition to seven general meetings that were held, there were held five priesthood meetings, at which the Elders reported their labors, and I gave them instructions, and counsel. Friday night, Nov. 30th I blessed Sister Nellie Hunters baby. She wished me to give the baby a name; I gave the little girl baby the name of Maurine name of our baby born while I was on this mission. Baby was eleven days of age when I blessed it.” (Duffin, 1900 Nov 30 Fri; The meeting was of the North Texas Conference, which included the regions around Dallas); “We ate breakfast. and after talking a while with Mr. Bass. we started back to see Bro Ben Bass. arriving about 10 O’Clock. We went from there to Mr Burnett. where we had dinner. After which we all went down to the Creek. Where I baptized Bro and Sister Burnett. and. Miss Lillian Bass. Then. we returned to the home of Bro Ben Bass. where there was. 11 of the Saints. and some. friends here we administered the Sacrament. and. spoke on the Word of Wisdom. and Tithing After Meeting we blessed 5 children three of Bro Burnetts and two of Bro Bass’s Then we started for Bro Barneycastle’s. arriving. about 6 O’Clock where we spent the night.” (Forsha, 1900 May 29 Tue)