No mulch, it smothers moss, which likes bare ground and good contact with soil to be happy. Just keep it weeded until the moss spreads. If you feel the need to put something in the bare spaces, you can always lightly scatter small river rock. That won't stop the moss from spreading. But moss doesn't grow on compost.

Moss that grows on wood generally doesn't do well on the ground, either. You might have better success with native mosses than those bought online. And paying attention to how and where they are growing in their natural habitat and trying to duplicate those same conditions in your garden is important. There are mosses that thrive in damp shade, and there are those that prefer dry shade and some grow in full sun. I do like to find mossy rocks and pieces of mossy wood and place them as is in my moss gardens. The moss will often spread off the rocks and pieces onto the ground. Some moss is very slow, some spreads quite fast and will often cover a sizeable area in a couple of seasons. I actually got some from a friend with a greenhouse, it grew rampant under the benches and he scraped it off each year. It grew as well on a mossy path at my house!

When you transplant it, take a good layer of the native soil with it if you can, clear a spot where you want it to grow, down to bare soil or even dig a small depression. Press it firmly to the ground, sometimes I even step on it to make sure it is tightly seated, and water it pretty well, not for the sake of the moss but to make sure it is stuck down.

You can be too good to moss by watering too much, also. Just a light watering once or twice a week is all you need, moss doesn't have roots but it does like damp soil.
Too much water will make it turn brown, and I think that the chemicals in city water will also. Rainwater is best. And moss does sometimes go dormant in the summer. It doesn't hurt it to let it go dry as it will recover with the next rain.Dog watering will also turn it brown!

I find that the best time to transplant moss is in winter, it does a lot of growing in cool weather and besides, the birds and squirrels aren't actively digging it up as fast as you plant it, as they are in spring. Tho that is not necessarily a bad thing as they scatter bits of moss everywhere, and even if you have to replace divots when they are dug up, the bits will spread and grow.

If you watch carefully, you can find moss growing in the cracks of sidewalks, sometimes in city parking lots and next to the sides of buildings, Occasionally you can rescue mosses from areas being cleared for building sites.

I've never tried the blender method, I've heard mixed reviews on it, but it might be worth a try. If you elect to purchase moss online, if you describe your garden area, I am sure the moss experts at the nursery can advise you about the correct moss for your climate.

Sandy