Doug White wrote:

The question is whether code requires any sort of bushing or clamp on
either end of the conduit.

Yes.

Loose wires run in conduit do not require
clamps. If the Romex was exposed where it came out of the conduit, a
clamp would be required that fits on the end of conduit. It would take a
much larger hole to fit the clamp down through the top of the wall, and
the Romex is never exposed anywhere.

There is a standard conduit fitting that goes on the end of the conduit
and provides a NM (Romex) type clamp. You run the NM cable up to the end
of the conduit, and strip the NM jacket off for the length that will be
in the conduit.

You will need to remover some of that wall paneling so that you can
access the interior of the wall at the point where the NM will
transition to EMT conduit. You can insert the conduit through a tightly
sized hole in the top of the wall, install the NM connector on the end
of the conduit from inside the wall, and then feed the wires from the NM
up through the conduit, finishing with an inch or two of the NM jacket
in the conduit and clamp the NM cable securely at the NM to EMT
connector.