This is a general answer which can be refined as more information becomes available. Your best bet is to make a timeline of events and associated records for all the individuals involved so you can use it as a reference.
When determining whether an individual is in any archival record collection, it helps to gather the following information:
- any and all names that the individual might be known by
- the dates of the relevant events you want to look for
- the places where the events took place
In her presentation from NARA's 2015 Virtual Genealogy Fair, "Broke, But Not Out of Luck: Exploring Bankruptcy Records for Genealogy Research", NARA archivist Jessica Hopkins described this trio of name + time + place the "three-legged stool". If you have all three elements, you have a stable place to stand when you begin your search. This is a sound principle for any research question that involves searching in an archive because:
- Records for events like immigration are kept because of laws. Knowing the date the event might have taken place is a pointer to what laws were in effect at the time and what laws were in effect. (Keep in mind that when laws change, there may be a lag time to get a new system in place, and there may not have been 100% compliance at first.)
- Knowing the place the event took place may determine what archive now holds the records that were kept, or if they are all in the same archive, they may determine what record series or fonds hold the records you are looking for.
- When you request the record, it helps to be able to give the archivist enough information that they know they're pulling the right file for you. (The same principle applies when accessing records online via a search form. You need to be flexible in the use of wildcards and keywords because you don't know how indexed records might be indexed.)
The FamilySearch Wiki article Tracing Immigrants Search Strategies is a sub-page under United States Emigration and Immigration, but the same principles apply. Start with the records your cluster created in the United States, then work backwards in time and search for Canadian records before trying to move back to France.
Study the entire family group (the F in FAN Club) and study all the records as a group as you analyze them for clues. Along with your timeline, make a log of the records you have collected, listed in the order the records were created. Putting the records in their own 'birth order' helps you keep track of which events (like birthplaces) are found in records created much later than the event happened.
A final caution: While it is important to keep the dates of immigration events in mind when you are thinking about what records may have been created at the time, bear in mind that other records about the event can be created much later on. For example, if a person naturalized in the United States after 1906, their petiton might talk about events that happened decades earlier, and if they corresponded with the US goverment about their naturalization, those letters might have been created decades after that event. The same principle applies for any event where an individual contacts a governmental agency. If you search too narrowly, you may miss out on finding relevant records.
Note: I have added links to Naturalization records in Canada because information about immigrant arrivals is often included in documents about an immigrant's naturalization.