Ableton's own Learn Live series walks through how to set up sidechain compression using Live's Compressor, with a classic dance music example: chords ducking every time the kick drum hits. The setup covers enabling the sidechain, selecting a source track, and drilling down to individual Drum Rack chains as the input.
You can tap into the chain signal before its effects, after its effects, or after its mixer, each giving a different level coming into the compressor. The Sidechain Listen button lets you monitor that input directly, and the built-in EQ lets you isolate specific frequencies, which is especially useful when your sidechain source is an audio clip rather than a routable chain.
Once routing is in place, Threshold and Ratio control how much compression gets triggered on each kick hit, while Attack and Release shape the timing of how the ducking fades in and out. Dialing these in is where the pumping effect goes from mechanical to musical.