The best way for you to get information is to call the school to which they will be moving. The most important part of this dilemna, to me, is how to do it so that your children will most likely be at the same educational level as the students at their new school.
If it were me, I'd keep them in their current school until the end of the school year or as close to the end as is possible. I think the states teach the same subjects during the same grades. You can check this out with the new school.
I think starting them at the new school just a few weeks before the end of school is not a good idea. The new school is probably not teaching the same things that they were learning at the old school because they are closer to the end of the year. Your kids would be at a definite academic advantage. They would miss a months worth of work. That would not be so much of a problem for the 6 yo. No problem at all for the 4 yo, but may be for the 10 yo.
Summer school doesn't necessarily teach the same curriculum that is taught during the school year. I don't think summer school would provide the knowledge that the end of the year would have provided. Summer school does increase skills in specific areas such as reading and math. If your kids are weak in any area, summer school would be good idea no matter your decision. Also, an advantage of summer school in the new district is that they would meet kids and begin to make friends.