Type of crane to move stone slabs?
An engine hoist with large wheels. If you can't find one pre-made "offroad engine hoist" will provide many design/modification ideas.

An engine hoist with small wheels and planks or plywood laid on the ground to make a hard surface to run on.

Same options as above but an A-frame hoist.

A 3-point hitch crane on a tractor.

A small excavator (rental, one day should do if you are prepared.)

A hitch-mounted crane on a pickup truck.