Clan Games is an event during which Clan members complete challenges to maximize the entire Clan's rewards. The Clan Games Caravan appears when Clan Games start. Town Hall level 6 or higher is required to participate.
A Clan has 50 slots for participants. When you assign a challenge for yourself, you enroll yourself in your current Clan's Clan Games. This uses up one participant slot.
The above could result in the message "Too many people in this Clan have already taken part in the Clan Games" to be shown. This means that 50 members have assigned a challenge to themselves, whether they are currently in the Clan or not.
After you choose a Challenge, it is not possible to leave the Clan to participate in another Clans' Clan Games.
If you are in a different Clan when Clan Games end, you will collect the Rewards from the Clan in which you completed at least one Challenge.
If you are kicked out of a Clan, your points still count towards a Clan's total points and you get the Rewards from this Clan.