I’m not an attorney, but have had experience similar to yours. Here are my thoughts… naturally this advice applies only after you’ve done all you can to negotiate out of court with your children’s mother.
You need to file a complaint for primary custody of the children, citing the information you provided previously. You will need to demonstrate that mom’s frequent relocation and interpersonal instability is detrimental to the children’s stability and emotional health. Sue for full custody, visitation for mom and child support.
You must be able to thoroughly establish with legally verifiable and admissible evidence 1) your strengths and superiority as a parent 2) mom’s weaknesses and inferiority as a parent 3) damage to your children’s emotional, mental or physical health as a result of mom’s instability - and how you can provide constancy, stability and appropriate ability to provide for the children’s needs. At the time you file your complaint you may wish to request a restraining order prohibiting mom from relocating the children again, pending further order of the court.
I’m curious… how are you doing a 50/50 split with her moving so frequently? Has she stayed in the same town/school district all this time?
Regardless - I cannot state with sufficient emphasis that custody litigation can be pure hell on everyone involved. If you get to a point where you absolutely must file for custody (which you very well may need to do), then take advantage of the mandatory mediation session to try to settle differences and come to an amicable agreement. Custody litigation can be very expensive and drawn out - to say nothing of the emotional havok that it inflicts on the parties involved.