Answer deleted. Sorry. I misread your post thinking you were referring to an ex-spouse. Thanks, @Critter.
When you file a joint return it doesn't matter where anyone lived .... you can claim the child care expenses on a form 2441 as long as both spouse have income on the return.
On a joint return only, “lived with you” means lived with you or your spouse.
When filing joint and claiming childcare expenses, which parent the child lived with doesn't matter. However, to qualify for those expenses both parents must have taxable income.
If only one parent has taxable income, the child care expenses are still allowed if the other parent was a full time student and the child care was paid so that one parent could work while the other attended school, or if that parent met one of the other requirements. The program will ask about that if both parents did not work.