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Mulberries should be pruned when they are dormant. They really do not need to be pruned except to shape the tree and clean out any weak or dead growth.

If it has massive leaf loss, it is likely due to either a fungus, pests or root damage. I am leaning towards a fungus as it is still producing fruit. Pests and root rot would wipe out the fruit but fungus can leave it intact. Treat the tree with a fungicide to treat any fungus on the tree.

Spring or early fall are the best times to transplant.

That being said, the issue is not so much that the tree is too young but rather that it may be too old. Plants where not intended to move around, so the longer a plant is in a spot, the harder time it has moving to a new spot. Yours can be moved, but it will have trouble and you will risk losing it by moving it. This article will help you reduce the chances the tree will die when moved:
https://www.gardeningknowhow.com/plant-problems/environmental/learn-how-to-avoid-and-repair-transplant-shock-in-plants.htm

Hi there anon of L.A. The Mulberry tree is a very invasive shallow rooting tree,so if you want a shade tree that keeps its roots to itself-choose something else.As I live in Australia I can't advise you of an alternative.P.S.The fruit are very tasty & can be frozen for up to 3 months but have a very short shelf life, that's.why you don't see them in s/markets Regards -Bob from Toodyay W.Australia

Did you have a frost in your area? New leaves on trees are very susceptible to frost and a frost can kill the leaves, but leave the tree healthy. The day after the frost, the leaves may have appeared fine and the damage, such as brown or blackened leaves, may not appear until 1-2 days later.

Another possibility is that the tree got overspray from a weedkiller. If someone was spraying something like Round-Up on a windy day, the spray could have drifted to the tree and may have cause leaf die off like that.

In both cases, the leaves should regrow without a problem.

It may be pests like aphids or it could be a fungus. Both can cause these symptoms. Treat the tree with neem oil. Neem oil treats both pests and fungus, so which ever it is, it will take care of it. Get up as high as you can get it. It is systemic, so will be absorbed by the tree and will help protect the whole tree.