I'm in agreement, something is probably clogging the filter. No fluid, no movement.
As for the mud, is it dark brown type mud? That's clutch material. Water in the trans fluid (oil) would appear milky reddish. FWIW typically you will get trans fluid into the coolant and not coolant into the trans fluid. This is because the trans fluid is pressurized higher than the radiator.
Likely a seal is not quite right and that caused some slippage and the clutches to wear quick, build up in the pan and starve the trans.
