Does a "Visa on Departure" exist?

(some of the info here might be dated – but still it might show that such a process is indeed recognized by some international laws)

Yes , Since I am travelling really a lot for business, I have encountered them in some (special) situation, although not in Airports, but in walk-in borders , trains or ferries – and usually where the countries have a special relation.

For example , in Zuhai (China) where there is a walking border to Macau , you can apply for a special 3 days Chinese visa at the border itself .
In Shenzen, You can apply for a special 5 days Chinese visa at the border also.
Now, while those situations Can be considered a “visa on arrival” – for Chinese citizens, it is the other way around .
A Chinese citizen can apply for a Macau visa (or respectively – HK) at the border itself BEFORE exiting China .
This can be also done for other non-Chinese citizens that require a visa to HK or Macau ( I do not require one – But I have seen it done there )

Another Example is the walk-in Border between Jordan and Israel when one can apply for a Jordanian Visa BEFORE exiting Israel (there are 3 such borders , I only have done it in one).

Another example are some remote borders (Can not remember the names) between Ecuador and Colombia and Ecuador and Peru, where some years back (actually – a lot) I had a visa made BEFORE exiting the Ecuadorian customs, and the visa was made by Colombian and Peruvian officials INSIDE Ecuador .

India and Nepal (were) another such example – and come to think of it – so does China and Tibet ..

Back to China, in Guangzhou South train station there is a special direct train to HK, where you actually do passport control at the train station (more than 150 km INSIDE china) before boarding the train. A few years back (maybe now also – I do not know) You could have applied for a visa there also (before buying a ticket)

So I guess yes , In some special situations these type of visas exists (or – existed) so they are sanctioned by international law and applied somewhere – although today in the age of e-visas and computers , I guess they are disappearing or will soon .

When flying from Mexico to Cuba, I bought my Cuban visa in the Cancún airport. The visa was “approved” by virtue of being paid for, but I still went through the normal immigration process in Havana, and presumably could have been denied entry at that time.

This sounds like it meets your criteria for visa on departure, but as the visa process was quite lax, it’s functionality more like a simple entry fee, which can be paid for “anywhere”.

In order to have a ‘visa on departure’ the destination country would need to have staff or a sub-contracter at every airport with the service offered. And the costs for such would be high and would only be cost effective at airport with high volume to the destination.

The USA has CBP agents at a few global airports handling pre-clearance for flights to the USA. But that is only offered at a handful of the numerous international airports with direct flights to US airports.

Visas on Arrival sometimes take time for all to be processed, which is not a huge inconvenience to flight operations. But processing visa requests before a flight could either delay departure or cause folks to miss their flight.

That is, a visa that is applied for (and granted) at the airport
before boarding or checking in for a flight to the visa issuing
country?

Not a visa per se, but there are examples of pre-authorization that is required for exiting a country.

In India and Pakistan (perhaps others, I know personally of these two countries) in addition to a normal visa, you also need a “cleared to leave” stamp which is given only at the airport by the immigration department. If you do not have this stamp, then you are not allowed to depart for the UAE – note, this is specifically for the UAE.

In Kuwait and other GCC countries, there is a specific “exit permit” which is a permission to exit the country; but this is to exit to any country and not just a specific country. It is given at the airport.

There is also a specific “exit visa” which is given for expats in Kuwait that allows them to finalize their formalities / documentation before exiting the country permanently.