Under the Uniform Interstate Family Support Act, states will enforce the issuing state's child support orders, including garnishment requests.
The employer must honor any support order they receive from another state. The issuing state will determine the amount to withhold, arrears, duration, where to remit, etc.
Connecticut refers to these as Income Withholding Orders.
If the child support enforcement division is involved (typically it is, for interstate cases) then they typically seek garnishment as a matter of convenience.
Rules for the issuing state here: http://www.ct.gov/dss/lib/dss/PDFs/WITHOLDING.1999.pdf
It sounds like you have already completed the exemption and modification request.
The court may suspend/withdraw the wage garnishment if they deem it is not necessary. http://ctlawhelp.org/guide-to-wage-attachments#
Normally an unpaid medical bill is not a reason for garnishment unless the parent had knowledge of the bill. Typically the court order will require the parent to notify the other parent within a certain time frame, so that the child's medical bills can be paid in a timely fashion. If that was not provided in the order, and there was no notice of the bill, then that generally is not a reason. It is also a good idea to "modify" the order so that there is the notice requirement for future medical bills.
If there is a history of ongoing payment without issue, the court has the discretion to order direct payment. If however, the child support division is involved, they will typically allow garnishment simply because it is an easier process for the department (for accounting purposes).
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