Do games purchased in one country work on an Xbox 360 which is bought in another country?

I have bought some old Xbox 360 games from one of my friends in Switzerland and I am still in Switzerland. I have an Xbox 360 which I bought around 8 years back in India. Will these games work on my Xbox which was bought in India?

If not, is there an alternative approach to play these games?

TL;DR The games will probably work just fine.

The XBox 360's region-locking has three 'regions': NTSC-U/C (The US and Canada), NTSC-J (Japan) and PAL (Most everywhere else). Since India and Switzerland should both fall in the PAL category, you ought to have no trouble using a game bought in one on a system bought in the other.

Now, if your friend in Switzerland imported these games from Japan or the US (which I doubt this person did, but I suppose you never know), the games may or may not work on your Xbox 360 because Microsoft allowed each individual game publisher to decide whether they wanted to region-lock their title. This answer on Arqade shows how to check the region of the game using the game case, so you could check real quick that way to make sure all your games' regions match.

(Additionally, Play-Asia has a page where you can check the region-compatibility of imported games, but if you have the game cases in front of you then just checking them manually is probably easier.)