Just adding our experience to the mix; 10/30/14;
Cozumel;
PP stamps at post office just one block north of the
DCL pier/Punta Langosta. West side of street, easy peasy. One person in front of me. Had to wait 30 seconds. Very nice attendant, ... gave me a choice of stamps....

In cab, 1 minute back to port. Walking would have been 5 minutes max.
Grand Cayman;
Office on south side, within secure pier area, as noted previously. Just ask. Great for stamps if you arrive in am/while ships are debarking/tendering. We were rushed and in a hurry so missed the "in house PP stamp" option. This south side office within the port area closes once the vessels are cleared. Meaning NOON.
Alternative PP stamp locale; Go out the secure gated port, head south, (right) the second or third building that is two stories is the immigration office. It is not labeled. IOW, the first two story building you come to is the Immigration office. In fact it is labeled something else. Enter said building from the front. , find stairs to go upstairs, bank left, say hi, get stamps.
Falmouth;
Unfortunately, regardless of the wonderful new port and security path configuration to potentially offer PP stamps easily, Falmouth refuses.
IMO, this would be the easiest port to do PP stamps, but NO ........
I told them they could charge for stamps, and pax would pay. They looked at me like I was an alien.
I followed up with an email to the Falmouth Port Authority regarding the desire of pax looking for PP stamps. I'll give it 3 months and they will offer PP stamps for $5 USD fwiw.
Castaway Cay;
Go to GS early, let them know you are there for PP stamps. They may take your PP away to get stamped. This event is dependent on the Bahamanian official. Sometimes they stamp PP, sometimes not. Still worth an early visit IMO. (as stated above)
HTH