Yes. FEMA did administer $640 million in grants from Customs and Border Protection in 2024 and about $364 million in 2023 to cities and counties that serve migrants. However, that program called the Shelter and Services Program, or SSP, is separate from FEMA’s disaster relief fund which is more than $20 billion.
So, Shelter and Services Program is a part of the regular FEMA budget. It's accounted for at the beginning of their fiscal year. Hurricane and other disasters aren't accounted for in the budget because no one can predict them.
When FEMA said, "hey, we're gonna need more money for this hurricane" that is normal. It happens with every large disaster.
You can read more about it directly from FEMA here: https://www.fema.gov/disaster/recover/rumor/hurricane-rumor-response