What should I do about my relatives who insist on playing musical cartoon channels for their children despite my advice to them?

It is not permissible to watch movies accompanied by music if the music cannot be avoided, whether these are children's movies or otherwise. The purpose of quieting children does not permit playing music; rather, they can be occupied with other permissible things. They must be raised to avoid listening to music and other forbidden things. If you explain the religious ruling in a good manner and they do not respond to you, then there is no blame upon you.