I'll edit the practice area to "family law." Are you representing yourself?
SAN DIEGO CRIMINAL DEFENSE ATTORNEY--20 years experience
It's up to the court. Usually after a few appearances and conferences (at least in New York), the court will set the schedule for trial and any Pretrial conferences, filing of witness lists and the like. Last minute motions are frowned upon. While both parties can and do often settle on stipulation, making a contested motion to dismiss another parents petition on a dispositive motion right before trial would be hard to imagine succeeding, unless the other parent has simply stopped showing up on court dates and the motion is granted by default. I find it hard to imagine what the basis for such a motion might be: a parent was just convicted for child abuse or a sex offense or murder during the pendency of the proceeding and was going to prison?
You can't dismiss a case ex parte. The other side has to have notice and an opportunity to be heard on your motion. The other side can, however, default by not showing up or answering, in which case your motion will be granted by default. You should speak to an attorney. One will usually be assigned to you if you cannot pay.
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