From what I can recall, the CBN association is perpetual, regardless of who owns the network.
Per the original Fox agreement in 97-98, the "Family" name and CBN programming must be aired on that channel, regardless of owner. Any attempts to dump the "Family" name or the CBN programs, kill that contract, and kill the "Family" channel, and ABC/Fox would have been forced to create an entirely new channel to replace it, and no cable company would be obligated to put the "new" channel in the existing "Family" channel slot. (Blame Pat Robertson on that one)
Per the original Fox agreement in 97-98, the "Family" name and CBN programming must be aired on that channel, regardless of owner. Any attempts to dump the "Family" name or the CBN programs, kill that contract, and kill the "Family" channel, and ABC/Fox would have been forced to create an entirely new channel to replace it, and no cable company would be obligated to put the "new" channel in the existing "Family" channel slot. (Blame Pat Robertson on that one)