he avoids opponents' blows by a combination of moving out of the way and lightly smacking their limb away with his hand
While the previous answers are totally correct in their statements about movie technique vs. real life, this observation actually strikes to the core of your understanding about the role of "blocks" in martial arts.
I'm going to approach this from a karate perspective, and I'm going to use three basic blocks as examples; soto uke, uchi uke and geidan barai.*
The first two mean respectively outside and inside block and are traditionally taught to block incoming punches. Personally I would never use them for that. Ditto for the third one which means groin or downward block and is traditionally taught to block incoming front kicks - to use this for that purpose is insane, you will end up breaking your ulna bone while smacking it against someone's fast moving shin bone.
So what's the point?
The point is that these are still valuable moves but they are not strictly for "blocking", rather they are for grappling and striking once you are correctly positioned to use them. Can you use them as blocks? Yes. Should you use them as blocks? Mostly no.
The real point is that you don't need traditional formalised "blocks" to deal with incoming techniques. Parrying or slapping away the approaching limb is a perfectly adequate defense provided you do something immediately afterwards - if you don't then you and the opponent are back to the start again.
For the sake of the movie Bruce Lee will dance around and parry away the first 10 or 20 attacks before doing something - in real life you will do something immediately after the parry; you will move in, change the angle, execute a strike or grapple. The parry is the setup for the next move.
*These example videos were selected simply because they were top of the search results, nothing more. Using them is not an endorsement of their quality or correctness.