If your tree is rootstocked, then you absolutely don't want to propagate suckers since they come from the rootstock which is likely of very poor quality fruiting habits. Don't do it!

It is rootstocked. I am trying to propagate the rootstock so I can graft on top of it.

I like to take a spade and sever the connection between the sucker and the tree. Once it is clear that the sucker is self-supporting (remains healthy and keeps or resumes growing) it is ready to transplant.

I'd do the severing during a time of year that it doesn't have water stress. In my area that is pretty much any time October through May. If you are irrigating and aren't in too hot and sunny a climate you could probably do it any time.

I'd do the transplant when it is dormant.

You can skip the pre-severing/recoup step if you are willing to take a little more risk.

Spring is a good time to transplant,..then graft the following spring or bud later the same year.

Or..you can graft onto the suckers next spring and transplant when dormant.

If it was my peach, I would dig it early in the spring while it was dormant. I would let it grow for a year to give it good feet before grafting. I might take more than one, and use subsequent examples for stooling (more clones).

Thanks all for the suggestions. So I'll wait until spring to transplant them. It looks like my peach tree has bacterial canker just this year, so I don't know what its outlook is. It is still young and this year will be the first for fruit.