As far as I know, this should be for free. Normally on the website of your Catatan Sipil you see the prices for the different services they offer.
Something else on moving etc.; half a year ago the Ministry of Internal/Home Affairs proudly announced that instead of four administrative steps, now it only takes one step to process a change. So you don't have to follow the path RT/RW/Kelurahan/Dukcapil anymore for a mutation, but you can just take a copy of the KK to the Dukcapil.
Well, you wish. In our Jakarta Kota they scoffed that they never received that instruction and that here we still had to follow the former full trajectory.