Would depend on how thick you were going to make the mud. As long as it's basically liquid, then drainage should be OK as long as you put plenty of water in. if I was doing it, to drain afterwards I'd use a shower attachment (but with the shower head unscrewed, so just the hose), and have the hose delivering cold water right at the plug hole, before I pulled the plug. That way you're diluting the mud as it flows into the drain.
Clay and silky smooth river mud, yes. Generic yard dirt, potting soil or peat, no. I'd avoid using anything with tiny bits of rock or twigs that can create blockages in the drainpipe. Also, we've started using one of these bathtub drain screens (avaiable at most hardware places that sell anything tub related) to ensure clumps of any kind get filtered out.
With anything regarding mud/gunge & pipes, dilution is the solution.
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