xAP is civilian liner and thus lacks loading/unloading equipment. If island has very small port (under 2..and even size 2 it can lake long time to get unloaded), ship is not able to unload some heavy equipment. Mainly these are radars, engineering vehicles, motorized support, observation corps stuff etc.
You'd need to bring in unit with Naval Support Squads to help unloading (or enlarge the port with engineers).
The auXiliary AP don't land them even in amphibious mode? Not that I noticed, but worth checking out. That could explain some 'weird' stuff I myself experienced.
xAK/xAP= civilian ship pressed to military service. They do not have loading/unloading equipment like AK/AP or integral landing crafts like AKA/APA.
It doesn't help much either way if ship is not capable or port not large enough for these ships. Only Japanese can do good amphibious landings with them during first 4 months due to Japanese landing bonus.
"To meaningless French Idealism, Liberty, Fraternity and Equality...we answer with German Realism, Infantry, Cavalry and Artillery" -Prince von Bülov, 1870-