The 'Babylonian New Years' trait actually has my Ruler and Heir locked in their respective cities for life, seemingly. It has been discussed in the bugs forum and is a major problem at present with the only fix being to remove it from the script as far as I am aware. One season I have no problem with however.

I agree with client rulers being able to move a little also. At present defending beseiged cities with level 4 governments is irritating, requiring a field army nearby to initiate the 'sally forth' unless you want to try and sit out the siege (bad idea unless heavily outnumbered). I now only use level 4's well from the frontlines where a governer is desired and there are too few FM's to help (such as remote provinces with goldmines and/or at risk of rebelling with no governer). There is a fix in the 'unofficial mods' section to allow this, though I understand it is a bit random in function at present.
Some movement would also be nice to check out what types of bodyguards they have, escape plague and to hire mercs, though the priority would be to allow sallying forth.