What if their father plays no significant role in their lives and doesn't care what I do with them? Thank you for replying!
When it comes to international relocations with children, "handshake agreements", "understandings", "father doesn't care" and all that is bs. What matters is what is in writing, often confirmed by a court. If you can't find him, there are procedures for that as well. If he wants to be a pain, than you have to go through that pain and get it sorted.
Whatever you do, do not run off with the kids without his permission. Doing that will put you into the world of the Hague Treaty on Child Abductions which can be rather complicated and difficult, for you.
Good luck.