When Mega Bank bought the bank our firm used, we moved our account. We wrote letters to all holders of outstanding checks giving them the date the account would be closed and asking them to negotiate their checks immediately. Most called back asking us to replace the original check, saying it was lost. We did, with a check drawn on the new account. Then, we closed the old account a few days after we said we would. (It does make more sense to me to ask them to cash the outstanding checks rather than hold them. Those they hold will have to be reissued.)
As John notes, some states assume fraudulent intent when a check is returned "account closed." How you go about closing an account with outstanding checks is important.
"It is curious that physical courage should be so common in the world and moral courage so rare.” ― Mark Twain