Have never tried moving Asparagus, but have successfully removed some and planted some that I grew from seed. Two or three years later in the ground and they produce wonderfully, but there is the year from seed to growth in a tray before planting. Found it to be about the same time to harvest as buying a new crown.
Been picking my older and newer ones for last few weeks.
I looked through many of my very old gardening books from the 40s/50s and 60s, they all say much the same thing. I will list their main points here from all the books I read, all generally agreeing;
Established plants from a few years old and more will have roots that are entwined together and with other nearby plants. The roots can be six foot down and 12 feet spread.
They all mention a poor success rate from lifting/moving that will most probably damage their root structure, unsettling the plants and slowing down their recovery, so a poor crop for a few years. Hence would be better to get new crowns.
The lifting/moving advice mentions;
Doing from the edge of the plot, slowly and patiently easing the soil and crown with a fork and your hands and taking as much soil with the crown as possible, so not breaking up the roots too much. Their entwining will mean some breakage will occur. they all generally recommended using an upturned bucket to transport each crown, so you can easily lay out on your new prepared mounds and not damage any further. Ensure the soil is warm, so have the new prepared area covered for a week or so before laying out the crowns.
It recommends a generous watering of the crown once layed out in its new home.
Notice they all recommend doing this in late winter and early Spring whilst they are in a dormant state and not yet producing spears. Some mentioned you will know when your spears normally appear. They mention not doing over the coldest winter months, the cold double hit will harm them during the move. Some mentioned it can be done late Autumn when the plants are dying back and turning.
Good luck