If their father is still a Michigan resident, it is very unlikely that the Michigan court will allow a change of jurisdiction .
My response to the questions asked does not create an attorney/client relationship between myself and the questioner.
You are correct in that Florida is now the children's home state and that you and the children are legally residing in Florida, since being given permission to change their domicile. I would register a certified copy of your Michigan Judgment with the county court in the county where you reside. You should consult with a family law attorney in Florida to review your options and explore the necessity of registering your Judgment at this time. If he desired to pursue a change of custody, or a reduction of support, the same would take place in a Florida court after the registration of the Judgment. If you sought an increase in support, then you would have to file such a request in Michigan. Since support is apparently paid through the Michigan system and the father lives in Michigan, the court would not relinquish jurisdiction, especially since jurisdiction is not subject to "transfer" as such.
Neil M. Colman
Mr. Colman is licensed to practice law in Michigan. The response herein is not legal advice and does not create an attorney/client relationship. The response is in the form of legal education and is intended to provide general information about the matter within the question. Oftentimes the question does not include significant and important facts and timelines that, if known, could significantly change the reply and make it unsuitable. Mr. Colman strongly advises the questioner to confer with an attorney in your state in order to ensure proper advice is received.