Ableton's Learn Live series walks through the Tempo Follower feature, which lets an external audio source take control of a Live project's tempo in real time. The example here uses a kick drum microphone pointed at a live drummer, syncing a running arpeggio to the drummer's pulse rather than a fixed project tempo.
Setup takes two steps: enable the Follow button in Preferences under Link/Tempo/MIDI, then select the correct audio input channel. Once the Follow button is engaged, Live adjusts the project tempo to match the incoming audio almost immediately.
Live handles gradual tempo shifts smoothly, but dramatic changes typically take a few bars for the algorithm to lock in. The Follow button can also be mapped to a MIDI controller or keyboard key, so you can toggle the feature on and off hands-free during a live set or rehearsal.