Legal Question in Family Law in Connecticut

Not directly. The non-custodial parent could make a motion to modify and seek to have his travel expenses factored in when calculating the child support. A lot would depend on his reason for moving (job posting, new job?) and distance (are we talking Texas or Vermont?). If the move was legitimate, what they sometimes do is subtract his travel expenses from his gross income to calculate the child support. For example, if he makes $50,000 per year (net) and based on that income his child support would be $300 per week and his travel expenses would be $5,000 per year, they'd recalculate his child support based on $45,000 per year net income, rather than $50,000 per year, thus reducing his child support to $270 per week, rather than $300. It's not a dollar for dollar reduction from the child support itself, and in no case will they make you pay his travel expenses directly.