This time period is outside my area of study, so I don't have many practical suggestions to find records for the 1790s, but perhaps this will help you get started.
The basic principles are to start from the known, to work outward from that to the unknown, and to make as complete a picture as you can, so that with each new historical record you find, you have enough information to determine whether this is the person you are researching, or something that belongs to another person with a similar name.
I would approach this problem in two different ways. First, I would collect all the information I had about Francois Dusnesmil with his wife Eliza Champion and 3 children, and any other members of his family. I would make a list of all the sources I had so far about this family. I would extract all the information about them and make a timeline. Then for each event I would isolate specific items that seemed promising for further study. I would collect all the information I could find about their stay in the colony and afterwards, in hope of finding clues that would allow me to work backwards.
You want to look for ANY information you can find about this family, from cradle to grave, including any siblings or other relations of Francois Dusnesmil and Eliza Champion. Cast a wide net first, because you can't know which record, history, or biography might hold the clue you need that will lead you to the answer to your question.
The second approach would be to gather material about the colony as a whole. Especially when the historical records are sparse, it helps greatly to study the friends, associates, and neighbors of the people you are researching. (Elizabeth Shown Mills calls this the FAN Principle -- see her website for the article Identity Problems & the FAN Principle.) Learning their stories may lead you to clues that you won't find in a direct search for your target family by name. This will also allow you to look at the records that you do have in the context of history.
I see that the French Azilum site's own website is just getting started, and while there are links in the sidebar for "Biographies", "Families", and "History", there is no material associated with those links. So where can your visitors go to find more information?
The Pennsylvania Historical & Museum Commission's article A French Asylum on the Susquehanna River republishes material from Norman B. Wilkinson, "A French Asylum on the Susquehanna River" Historic Pennsylvania Leaflet No. 11 (Harrisburg: Pennsylvania Historical and Museum Commission, 1991).
The article Nobles in the Wilderness: The Story of French Azilum
By Nicholas Mattise, Fall 2009 includes a list of sources. Any bibliography or source list is useful not only as a list of what publications about the colony exist, but also for the pointers to repositories that hold these articles, which might contain other material.
Thus while the Wikipedia article doesn't have much in the way of source material yet, the list of categories for the articles has "French-American culture in Pennsylvania" -- a possible library subject heading for works about the colony.
The Family History Library has in their catalog a typescript The story of some French refugees and their Azilum, 1793-1800 by Louise Welles Murray -- the catalog entry has the subjects "United States, Pennsylvania, Bradford, Asylum -- History" and "United States, Pennsylvania, Bradford, Asylum -- Biography." and searching those subjects will give search results for other materials.
Other possible places for information about the colony might be:
For what genealogical records are available for the area, see
and similar guides to Genealogical Research in Pennsylvania. Don't forget your local public libraries or historical societies; they often have finding aids on topics of historical interest in the area. They may also have contact information for professional genealogists in the area.
Given that there aren't going to be as many opportunities to find records for 1790 as there would be in, say, 1890, I would look for any accounts about how people arrived at the colony to get a sense of the migration pattern. Once you have the bigger picture worked out, it may be easier to research the more specific question of how and when this family might have arrived there, and where they could have been before they arrived.
After that, follow the records. Always ask what records might have been created, and then look for which records might have survived and which repository might hold them.
