Shotgunlover. I have not as yet found a truly non staining adhesive though there are some that do come close.
COPYDEX Is a form of Latex emulsion that can be used as a straight forward adhesive or as a contact adhesive. It is as you would expect water based recommended for paper fabrics and leather though once dry it can be extremely difficult to remove in fact can be impossible though while still wet it can be removed with a damp cloth. Its strength is good for large flat areas but on corners I prefer to use an out and out contact adhesive though it is good for small repairs on vintage cases as well as a complete re-cloth.
EVO-STICK. Is a high strength contact adhesive for just about anything, it works well on paper fabrics and leather and does not stain if you remove it with its correct solvent soon after using. The down side it is Solvent based and in a closed room it can give you a bad time with dizziness headaches. It is excellent for patching and I have used it for a complete re-cloth.
Rabbit skin glue. Natural works well clean none staining if removed quickly with warm water, good grip once cool but you do need a double boiler glue pot to use it. Though the only Adhesive to use for a true vintage repair and remarkably it is still easy to obtain here.
All the above Adhesives are obtainable here in Brit land without to much trouble and I am sure COPYDEX and EVO-STICK equivalents are available in the US.