
Puzzles can be so pretty. Any clear glue will work and if water soluble so much the better for thinning. puzzle glue is ussally mod podge which is cheaper but the white elmers type glue is even cheaper.
I like Sandies idea for mounting them. If they weren't huge you could take 2 pcs. cardboard cut to the size of the puzzle put one on top of the finished puzzle which has the other pc of cardboard under it(put puzzle together on one). Carefully hold the two cardboards tight as you turn the puzzle over ,now it it put together and turned upside down to add a backing of your choice. After dry you can either flip it back over and seal the front or mount in a frame. This is easier than sliding a glued surface under a completed puzzle I think.