Here are a few things I found on the internet.
Each parent is supposed to pay for child support according to his or her ability and "circumstances and station in life". A parent with the higher standard of living has the obligation to ensure his or her children share in that lifestyle. A non-custodial parent cannot be forced to pay child support beyond his or her means simply to match the custodial parent's new "station in life" (as where custodial parent remarries into wealthier social position).
An order for child support can be changed or modified any time there is a "material change in circumstances" from the time that the existing child support was issued. A material change in circumstances can take many forms. The changes can be the result of changes in the parent's financial situation - such as appreciable difference in the amount of income earned, loss of a job, a large inheritance, or a change in the amount of time spent with the child. The material change in circumstance can be the result of a new situation for the child - such as large medical expenses, need for special education, or other unexpected requirements. A child support payment could be modified by stipulation between the parents ( as long as guideline support factors have been accounted for) or by a noticed court hearing.
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There was no material change in circumstances and as far as Item number 1, it is self explanatory--a non-custodial parent cannot be forced to pay child support beyond his or her means simply to match the custodial parent's new "station in life"
So taking all this into consideration can they go and order support just because they moved.