There's a Polish saying jak pies je to nie szczeka, which (literally translated) means "when dog is eating, it is not barking."
What is the English saying with the same meaning, used to mean that if we involve someone in other activities, he/she will stop what he/she is usually doing?
I have a half-answer. I have seen that phrase (or one very similar) used, but not to mean "if we involve someone in other activities, he/she will stop what he/she is usually doing".
I have seen it used to mean, "Don't talk with your mouth full." And there is really no other way of saying it that I know of.
One example source (Word doc; see page 8):
Gdy pies je, to nie szczeka, bo mu miska ucieka
Word-for-word translation: When the dog eats, he doesn’t bark, or his food will run away. Meaning: Don’t talk with your mouth full.
To provide an answer for the phrase meaning involving someone in other activities so they stop what they're doing part
A red herring is a distraction. Here's a better definition.
Also, sending someone on a wild goose chase for the purpose of distracting them (e.g. the police from noticing what you're doing) is a common theme in literature or Hollywood; though the definition on it's own isn't quite the same, it can often give that implication.
