No matter which substrate you want to use, getting a fishless cycle going before you move and including some of that substrate in the bucket is a good idea. I would also include some filter media, sponge, bio noodles or whatever.
Do not bother making the substrate layer too deep, the bacteria will not grow in the lower area, not enough oxygen.
When you set up the new tank put the rest of the substrate in first and use the cycled media from the fishless cycle on top. Again, the bacteria will not survive deep under the substrate. Depending on the size of the particles about 1/4" to 1/2" deep is about the max.
If you start with no source of bacteria the fishless cycle will take 3 weeks.
If you do a few water changes and add the mulm and dirty water to the bucket there is a small bacteria population in that, but not very much.
If you add some of the media from a cycled filter that is the best source of bacteria.
Read about the fishless cycle. To grow bacteria you need to feed them.