You would follow eviction practices and change the locks if they do not move out willingly. A soft landing would be to help them with first and last months rent but not co-sign an apartment for them, but it is not at all necessary.
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Not in and of itself, however is the parent has a disability that prevents them from being able to care for the child, than it becomes an issue of safety.
Unless there are unusually circumstances, once a minor becomes an adult under the laws of the state in which he or she lives (or as stated in the support order) child support ends. If he or she has established residence outside the custodial parent's home with the approval of the custodial parent, the obligated parent should petition the court to have the support order amended or rescinded. Child support money is for the support of minor children and not "owed" to the custodial parent. The custodial parent does have the legal right to sue the obligated parent for any arrearages or "extra" expenses incurred while the minor child was in his or her care.
Interpretive as any obligation to support runs until age 21
Not ordinarily, because the child becomes an adult at 18 and can make whatever decisions he or she wishes. There are some cases (notably legal incompetency) where the child does not gain free status as an adult, and would continue under the supervision of a parent or guardian.
Legally, if the child is still a minor than no. If the child is underage and leaves the home the parent could be charged with abandonment. It is the parents reponsibility to look after the child until they reach adulthood. What the child wants does not really matter.