The plane's future position can be accurately estimated as soon as it spools-up on the runway. So the drones wouldn't need to catch up, just fly up to intercept. To ensure turbine destruction, the drones could have a payload of ceramic ball bearings, explosives, or both.

A passenger plane taking off is going to have a hard time seeing (let alone avoiding) such small, maneuverable craft.

Of course, this is a movie plot threat. Really, nobody is going to spend the time and resources to program drones like that.