Depends on the type of residency permit you have, and, whether you travel by road, or air.
Considering that we are talking about SV Pilots, the answer is that you dont need ANY LETTER OF AUTHORITY for you nor your family to travel. The cities of Madinah and Makkah are out of limits for non-Moslems, but everything else is ok.
411A, in your day, a letter granting permission was required to travel more than 25 miles from your domicile, this no longer applies.
Lessnessman, all crews in Jeddah, but with rotations to Riyadh for a number of days at a time. i.e, day 1 depart Jeddah, 4 legs to land Riyadh, remain over night, day 2/3 = 4 legs to land Riyadh, day 4 = 4 legs to land Jeddah, or position as passenger on day 1 or 4.
Mutt