What is required by and for flights that cross borders in terms of border control?

Question

This can apply to countries that have water in between, like the UK-France or US-Canada.

1. If an aircraft crosses into neighbouring country, does it need to go to a border-controlled airport (with flight plan and access granted)?

An IFR-filed flight with permission from that country's civil authorities, but to a non-international airport.

2. Are international airports the only border-controlled airports or do some countries have predetermined airports that pilots can fly to, so that border control authorities can process them at said airport?

Answer 1

A famous US/Canada example is Niagara Falls.

As as student, part of my cross-country test involved flying a circuit around the Falls, which meant passing the international border. No special permission was required.

If I'd gone anywhere near the American airport (top right of map), the situation would have been completely different though.

Note that even though there are no specific border control related rules, there are many safety regulations controlling this circuit. (It's a lot more crowded now than when I flew it.)

GA Flying over Niagara Falls – Fly 'n Things

Answer 2

The details will vary by country, but in general, if you cross a border, you will need to get permission from one or both countries, file a flight plan and be talking to ATC/FIS as you cross. If you want to land in another country, you will need to do so at a designated port of entry. In some countries, you must also leave from a port of entry.

The term "international airport" doesn't have any specific legal meaning but is generally used by airports that have (or had, or hope to have) scheduled commercial flights to/from other countries. Many smaller airports are ports of entry too, either full time, part time or by appointment.