Is it permissible to extend one's hand to a child passing in front of a worshipper to prevent him from passing or sitting on the prayer rug during prayer, and does that invalidate the prayer?

The passing or sitting of children in front of a worshipper does not invalidate the prayer. They should be prevented from passing, but if they do pass, there is no harm, as the Prophet, peace and blessings be upon him, prayed, and children passed in front of him. The worshipper should have taken a sutrah.