I have not done this, but from what I have seen, the same way you go to another kingdom during an event.

When you click on a candidate empty spot, up comes an option to switch kingdoms, and the criteria that must be satisfied to do so. e.g. Number of carriages required in that instance.

Much like what happens when you try to go to another kingdom when you have troops bunking with you, or your troops bunked with someone else.

(Or trying to change profiles / gear / inquisitors, when they are out on a march or yielding?)