What is the legal process for kicking an adult child out of the house in Florida?

I've moved your question to the Real Estate Law forum since the process for removing a resident is the same whether the resident is a family member or boarder or what.

Since I assume there is no lease, the person is considered a periodic tenant. Without a rent payment this default to a month to month period. Florida law requires a 14 day termination/vacate notice. So the first step is to give them written notice that they need to vacate within 14 days (you can give longer if you wish). If they refuse to leave within the deadline, then you need to go to your local housing court for an eviction order. This will involve a hearing, but unless the person can provide reason they shouldn't be evicted an order will be issued. If they still refuse to leave, a sheriff can be hired to physically remove them.

By the way, you cannot change the locks (at least until after they have vacated).

Agree with Scott, you can not "kick" them out, you may evict them though legal process.