But first they have to convict the person. And they can't convict someone who is not an officer of the United States Government. The Senate can't impeach your mailman, your barber, next door neighbor or any other random citizen. Once Trump leaves office, it become moot.
There is precedent for a person who had resigned and was no longer an officer. The fact that he is an officer when impeached is enough of a hook. Trial later. Of course the removal part is then moot, but Congress has the power to taint. What good is the power if they don't use it?
"President US Grant's secretary of war, William W. Belkamp escaped conviction in 1876 by resigning two hours before the House of Representatives impeached him." The Senate did conduct the trial.