As far as the ACA goes, your ex is responsible for health care for any dependents he claims, or could claim, on his tax return. Same goes for you. In the years you claim the children, you are responsible for obtaining insurance. In these cases, the actual Premium Tax Credit paid will depend on the income on the return claiming the children.
So, if your ex is claiming the children as dependents, he needs to reconcile the payments. He would also be responsible for providing insurance or paying penalties in the years he claims the children.
If you and your children are all on the same policy and the same 1095-A in a year that you do not claim them, you need to follow the procedures for a shared policy allocation. (In TurboTax, you click on "I shared this policy with someone not on my tax return" when done with the 1095-A. In essence, you will claim only your own and your ex will claim the children's.
Under multiple support agreements, the parent claiming deductions for health insurance is dependent on a number of issues and how the agreement is spelled out. Someone else may be able to help you with that one.
