Well . . . it depends. You do not mention adoptions, so i presume no such thing was done (if so, the kids are treated precisely as they would if they were both parties' biological children). Technically, no automatic right to custody, or visitation, and no obligation for support.
That said, it is possible to have either continuing visitation, or possible support, in this situation. A great deal depends on the factual history, and what the adults (AND children) WANT to have happen; a full consultation with qualified counsel is a very good idea. For background, however, the relevant statutes, cases, etc., are set out at:
Sign up to receive a 10-part series of useful information and legal advice about the divorce process.