A few years ago, Oakland, California wanted to dredge and dispose of mud from their harbor. There was a lot of controversy because dumping it in the ocean was (is?) polluting fishing grounds. It will become a more frequent problem as we build bigger ships. But this is mainly because the mud was dumped from the ocean surface, mixing with water before it reached the bottom. (The San Francisco Bay is already too polluted to worry much about the pollution caused by dredging.)
Let's redesign the process from the beginning. We start with a combination scoop/barge which is dragged along the bottom to fill it with mud. This inherently takes less energy, since the mud isn't lifted out of the water and dropped into the barge. It would be pulled by a winch on shore, on a firmly anchored boat, or two two scoops would work together, pulling against each other. When a scoop, or two, are full of mud, the front door would be closed by compressed air, from the surface, or from tanks on the barge, converting the scoop to a barge. The same air would then be released into large, open-bottomed tanks, replacing water, to float the barge to the surface. The barge is then towed to the dump site.
As it approaches the dump site, some air is released and the barge is allowed to sink slowly. When it approaches the bottom, indicated by sonar or by a drop line, air tanks on one side are emptied and others on the bottom are filled, upsetting the barge. The mud falls out and the barge rises to the surface. Air is moved to bring it upright and it's returned to the dredge site.
Sinking, dumping and rising of the barge could easily be handled by automation. These might take some time, so it would be faster to use barges in relays - release full barges and pick up barges just emptied.
An earth-moving, if not earth-shaking, problem.