Mugo is probably the easiest pine to transplant - next to jack pine. It doesn't grow a tap root and should have fibrous rootage near the trunk. How you treat it kind of depends on how long it's been sited where it is. If it's more than 3 or 4 years, you might want to do it in two stages by root-pruning this fall & lifting it next spring. If you want directions on how to do that, I can help - let me know.

If you're to lift it (you didn't say where you live) this year, do it as most of the trees and shrubs in your area have already started to leaf out. Use a sharp spade to sever roots and get as large a root-ball as you can manage. Try to keep soil around roots as stable as you can. Move it onto a tarp & drag it to your already prepared hole where you will plant it slightly higher than previously (never shallower & allow for some settling of soil). Backfill with the same soil you removed from the hole & water in well. This plant grows best in a well drained and spare soil.

Al

I’ve got a mugo I’d also like to move but mine is maybe 4‘ tall x 4‘ wide and grown into some landscaping riverrocks.

How do I root prune?

You dig straight down, in the fall, all around it about a foot or so from the plant itself. That encourages new root growth do that when you move it in the spring, it will have extra roots and won't be as shocking to it when you do move it.