Same policeman, usually, however
OTOH if Immigration cared, why does anybody have EXIT controls? No country (other than closed dictatorships like e.g. Czechoslovakia used to be pre-89) should care who leaves it.
The UK doesn't for the most part care about exiting, which makes sense, but the rest of Europe certainly does.
The rest of the world generally does too,
AFAIK. Sometimes they have good reasons e.g. anti smuggling of artefacts (Egypt, Turkey, etc).
OTOH, if (speaking purely for the sake of argument and not suggesting anybody does this, and if they do it they do it not too often
) you depart a non-Customs airport on the Continent on a flight plan
filed for Le Touquet, with Lydd as the
filed alternate, and when you get to near LTQ you divert to Lydd, nobody is going to give a damn, and this method has been used for more years than anybody can remember. The diversion could have been due weather / sick passenger / etc, which is within Captain's privileges, and nobody in the system is going to bother to get historical wx and try to work out if there really was any bad wx in the area... I've done it more than once but only in cases where I was being blatently screwed around for extra money and delays ("due to a non EU destination") at the departure airport which had Customs anyway, and I overflew several other countries before announcing the diversion. I am not sure I would want to try it in the LTQ/Lydd case because it is so obvious if the departure was a small strip in France.