Please hire an attorney if you don't have one. You need evidence & witnesses to show to the judge what you are claiming. No judge wants a person like you describe to have a baby. Please protect this child! Good luck!
If CPS placed the child with you under a safety plan, the mother's access to the child should already be limited. It seems unlikely that the court will be moved just because the mother moved and wants it changed. Judging from the conduct that you referenced in your question, it doesn't sound like the mother has a good chance of getting the child back anyway. CPS will have to continue to be involved. It sounds like you already have custody of the child, the determination will center around what the safety plan states from CPS. For you to get custody and challenge mother's parental rights, mother needed to have left child in your care for 6 months for you to have standing to try and get custody. The main consideration for any case involving a child is the "best interests of the child," so focus on that. I recommend consulting an attorney to better explain the facts surrounding your case.
The mother should not be able to transfer the case because transfers are based on the location where the CHILD lives, not a non-custodial parent.
Before you have standing to petition for managing Conservatorship you must have had "care and control" of the child at least 6 months.
Contact the CPS caseworker & let them know what she is up to.
Consult with a family law attorney in your local area who is active in the practice of CPS related cases law. And bless you for protecting this child!
This answer is general information which does not establish any attorney-client relationship between the person asking a question and the person answering, or a duty to respond to ongoing questions; nor is it intended to replace competent legal assistance in the jurisdiction where the matter/issue arises or is before a Court.
Hire a family law attorney immediately! Since the child has resided with you for more than 6 months you have standing to petition the court for custody of the child. If everything you say is true and you have the evidence, you have a great chance of maintaining custody and limiting mother's access to supervised visits. I would also consider terminating her rights. Good luck!