My ex-husband moved my daughter to another state in August. I have moved to the same state (roughly a month ago). Should the CS case be moved to the same state or should it be left in the state it was started in?

As of right now the old state retains jurisdiction until it changes. If there is a dispute or modification then it can take place in the state you are currently in after you meet the residency requirements. ( usually 6 months)

Agreed, it will remain where it is, until there is a need to change it or change something in it.

ANY question on law needs to include your general locale as laws vary by area.

But generally, the state that issued the original order maintains jurisdiction until all parties have established residency outside the jurisdiction.