I have troubles finding the flight controllers recommended in the documentation on retailers easily available in EU, such as hobbyking . On the other hand, I find many flight controllers with the usual STM32 microcontroller but from unknown (at least to me) brands, such as this one or this one.
How can I know if they are compatible with Ardupilot?
If they are on the list they are known to be compatible. If they are not on the list… they will probably not work.
I look at the manufacturers website for a list of local distributors.
For example: the Kakute flight controller, the manufacturer (Holybro) has a list of distributors in the EU on their website https://shop.holybro.com/art/distributors_a0050.html
When you say compatible what exactly do you mean? The main flight code for ArduPilot is written in C++. Support tools are written in a variety of languages, most commonly in python. MAVlink is used for communication.
here is list of commonly used companion computers https://ardupilot.org/dev/docs/companion-computers.html
I mean that I can flash the Ardupilot software on the flight controller using Mission planner or some other easy-to-use toolchain. I don´t really know exactly if this is the case for every STM32-based flight controller. Thank you for the list of companion computers, but I am referring here only to the flight controller 
I look at the manufacturers website for a list of local distributors.
Thank you for the advice
Where do you live? I can hekp you find a supplier.
Which controller?
What are you building?
Thank you for that. Italy.
Which controller… that’s what I’m trying to figure out. Anything that supports a quadcopter, I guess. I am not looking for performance - just something cheap that can run the last version of arducopter.
I’m building a quadcopter on a f450 frame. At the moment I am using an old APM and I’m looking for an upgrade. The idea is to then test some Dronekit-based code running on a raspberry pi on-board connected to the fc via Mavlink.
Cheap:
Matek F405
Omnibus F4 Pro
A few more $$:
Kakute F7
Alright, thanks. I also found OpenPilot to be very cheap. Is it worth a shot?
Also, are these small boards losing anything with respect to a big bulky pixhawk except for connectivity?
The autopilot board inside a Cube from ProfiCNC isn’t much bigger than these smaller autopilots. The carrier board and redundant sensors is what makes the Cube bigger and more powerful.
Don’t bother with anything not currently in the Ardupiulot repository unless these things are satisfied:
The reason there are not more boards supported is typically due to 2 and/or 3.
The Cube is a good choice in Flight Controllers as are a few sourced from Mrobotics and CUAV. But they don’t meet my definition of cheap (<$50).
And of course these boards lose capability that’s why they are cheap. Redundancy is a major factor in terms of multiple IMU’s, Power strategy, and signal processing.
What is your budget?