Since your children are only 6 I would, in this specific case, definitely go for changing them into the local school. I'm not a huge fan of the Swiss schoolsystem but I feel that in cases like yours they are actually quite good and do most things right.

If you choose the local school you will ensure that you get all the support needed as it is the obligation of the Schulgemeinde to cater for all children, also those with problems.

As was suggested, I would _not_ inform the new school about the reason why you change into the local system. Just hint of change of circumstances or something. Let them start fresh in the new school. They will be evaluated together will all the other children as a normal part of the school day. If there are problems, the new school will discover them and they will then suggest the best course of action for your children and give you the support you need.

Good luck!