Much of the position updating and velocity correction ability of an INS is determined by the firmware/software in the system. That also determines whether a manual "position update" or re-alignment can be accomplished in flight. For example, the LTN-92s in current 747 Classics cannot be re-aligned in flight, and have no effective manual position updating capability.
The closer the Initial Position is to reality when the INS is aligned, the better it will navigate. While I don't know how far off the IP has to be to adversely affect one of these newer Ring Laser Gyro INS systems, some of the old spinning gyro types would start building stray velocities (nav error) if the IP was more than about 1/2 mile off.
AFAIK, the newer INS systems can be dynamically updated using GPS or VOR/ILS/DME, but the update applied by the radio nav instrument will "fade away" after the update signal is lost. Our LTN-92s will occasionally drift as much as 3 miles crossing the Atlantic or Pacific without updates (measured by the track error when updates are regained on the other side), even when there was "zero error" while updating at the start. Triple-Mix helps to smooth out these errors.