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Yes, the act does apply and the military member can request that there is a stay. However, there is not a right to the stay. The court simply has to take the member's service into consideration. The court can still take the children's welfare into consideration and, on that basis, push forward with a custody determination.
If the soldier is not deployed and has leave time available to him, the court can refuse to stay the hearings on the basis that his service is not materially effecting his ability to show up. The court can also make a temporary ruling while the case is stayed.
The court can, on it's own motion, stay the case or respond to a stay request from the soldier.