My ex and I divorced in 2003 and have shared custody of 3 kids. The divorce decree stated I would pay her child support since the kids were primarily living with her.
In 2005, my three kids moved in with me and I stopped paying their mother child support, but never had the divorce decree changed.
Today, I was served papers demanding back child support.

Is there a judge anywhere pm earth that would make me pay back child support to their mother when they were not living with her??

My advice is: Hire a good lawyer.

Can you prove that there was a change in custody? Was the change done through the court?

Is there a judge that would do that.. yes. So now because you didn't do it right in the first place you have a mess to clean up. You need to file for custody ( as it stands now ) and see what they say about the child support.

I always tell people that when you have a child support order it's like the meter on a taxi that keeps running, even if you are not in the cab, and the taxi is just waiting for you on the curb. It doesn't stop running until you make it stop.

Your child support order doesn't stop until the court makes a new order changing the old one. So, the kids came to live with you and you naturally stopped paying support? Great. Who cares? How's the court suppose to know that that happened until you tell them and ask that the order stop? The court's not going to know it and the order will keep running. An order is an order is an order. It's as simple as that.

Now, having said all that, in California at least (maybe elsewhere) you do have a defense to enforcement of the child support arears because of the de facto change of custody. It's based upon a case named Trainotti and it's still good law. The idea is that the child support was paid, in a sense, because while the kids were with you the majority of time you supported them financially and discharged your duty of support that came due under the order. It's an equitable defense to enforcement of the arrears that case law has carved out. But you better have proof of the change of custody--school records, witnesses, etc.

Other states may have similar equitable defenses so if you are not in California, check with a local family law attorney.