Well, from the research that I have done on Asparagus,it is best to transplant the plants in the Spring or Fall. Probably the spring would be the best as the plants will be stressed and a frost or freeze could kill them.
As far as transplanting them: Make sure that you build little mounds in the bed that you are moving the plants to. Lay the roots of the Asparagus over the mounds such that they are nice and spread out. The big mistake that most people make when transplanting Asparagus is that they leave the roots all bundled up. This kills the transplanted plant.

I agree about root spreading, but even wild asparagus has very long roots if its well established. The smallest plants should be dug only, as bigger ones will have lost their most important roots that can travel down over 10 feet or more. You mentioned 'going to seed'. If these wild ones produce small pea sized berry pods, these can be the best way to grow them. A few of my plants are female and these produce a few seeds every fall. I harvest these green pea sized pods and allow them to dry out a month or two indoors. Then, I pop the the pods open and remove the small balck seeds. These get further drying, and by January, I plant the seeds singally in individual 3-4 inch plastic pots with good quality seed starting medium. Once they germinate (up to 30 days or more), they start sending up new, tiny, spears. By the time early summer arrives, they are about a foot tall and are healthy ferns. These get planted out in my expanding asparagus patch. With these new plants being as small as they are, they still have lots of long thin roots which look root bound in the small pots. I spread these roots out a bit by digging a hole, and building up a pointed cone shape soil mound in the center of the hole. This allows the roots to spread out in a wider radius. Plant them about 3-4 inches deep, measuring from the surface of the crowns They will grow very well, and by fall, will usually have 2-3 foot tall ferns. Allow the ferns to remain until they die back, then cut them off the soil suirface. The following year, add a bit more rich soil or compost to the surface, as these also tend to expand and get exposed to more of the crown of the plants. Adding bone meal during the summer will help stimulate rooting. My patch as expanded three times now, and is of a size that is suitable for me to harvest several meals of asparagus. Every year, the crop gets a bit bigger. Have patience, they take several years to produce good sized spears. Wild ones may never reach the sizes of store bought, but would probably be much more tasty.

We have clay soil that is great for pottery making but absolutely deadly for gardening. Asparagus absolutely loves manure. You can pile it on top and it will kill the weeds but not the asparagus. I recently needed to move my asparagus bed so we brought in an endloader and just scooped it out. Then I pulled out the roots and reburied them in a mix of leaf mold (obtained from the city from their cleaning out of ditches) and horse manure (obtained from the county fair grounds horse barns). This was at the end of August. The asparagus came right back up and kept growing until a freeze.

Here is a link that might be useful: Blue Rock Station

I bought property in southernmost GA about 50 miles from home and planted 70 crowns 3-4 years ago. They did great, but now I just can't get there enough to harvest enough and to weed the beds. So I want to transplant some of them and grow them at home, perhaps in large containers.

What I don't understand is why people pull up the roots and start over again by spreading the crown over the mound in a trough. Why not just dig very deep all around the root mass without disturbing it and putting the large ball of soil in a big pot? Wouldn't that save you from starting all over and waiting another 3 years?