Yes, after all that is what you do when you buy crowns, although whther it is worth transplanting old plants I don't know (anyone know the average lifespan of an asparagus planT?). As with most things the best time is in winter when the plant is dormant, and you may have to wait a couple of years for it to settle down before you can crop it heavily.
I helped a friend move his established asparagus some years ago when he vacated his allotment. We thought it might be worth a try, otherwise they would have been destroyed anyway for the subsequent building work. Established crowns consist of a huge mass of fleshy roots which are brittle and difficult to dig up intact. They did not settle into their new site. He gave them three years before taking them out. Half of the crowns died, the others just sulked. Asparagus is of course transplanted in order to create a bed, but these are young plants which are adaptable and malleable, and certainly much easier to dig up intact. On reflection, it would have been a better strategy to have planted young plants rather than take a chance on transplanting the old crowns: the time spent witnessing the old crowns perish could have been spent watching the young plants flourish to reach harvestable size.