Other than a "dated" look, the old asbestos siding is good stuff. Paints well and is very durable. IMHO it is no better of worse than the new fiber cement fad that were into now, in fact I believe its less prone to water damage than f/c. Vinyl can be installed directly over the asbestos but you might want to take this opportunity to insulate with some ridged foam at this point. 60 years ago insulation was not high on the priority list. If the existing siding is "shadowed" by installing it on a lath strip (for a bigger shadow), you should at least install the 1/4" fan fold insulation to sort of hold the asbestos in place as it will crack (when shadowed) with the new nails from the vinyl, and pieces will fall down, not far but enough to show a bulge in the vinyl. I understand that this type of asbestos is NOT the pipe insulation that all the lawyers are chasing down victims of and it was taken by the landfill 'bout 10 years ago without any problems. This may have changed by now.
Depending on your wall insulation, think about 3/4 or an inch of foam (not the coffee cup stuff) if you can without creating trim or water problems.