Typically, each movement animation indexes through one “cycle” of movement. Animations are typically expressed as rotation, and perhaps translation, of “bones” onto which the “body” is rigged. When you move/bend a “bone,” the vertices and textures for the body part bound to that bone, move accordingly.
If you half-run, half-walk, both the “walk” and “run” animations are played, cycled through at the same relative speed, so they cycle at the same time.
Then, the joint bend values are blended between the two – the “left knee” might bend 22 degrees for the walk cycle, but 38 degrees for the run cycle, so in the “blended” animation case, if the weights are 50/50, the left knee will bend 30 degrees at that point.
Once you have the correct pose for each bone, the vertices are put onto the bones using a “skinning” process. This happens for each character, for each frame that's rendered.
The cool thing with this is that you can let other things bend the joints between the bones, too. If a weapon would stick into a wall, the game can detect this with collision detection, and bend up the arms so the weapon is moved out of the way, for example.
Certain blend modes may say that particular bones (say, the arms and fingers) are driven by some particular animation (reloading,) whereas other bones (say, the legs and torso) are driven by running and turning. Exactly how all of this is blended together, depends on the game engine, and the particular game – good engines have quite elaborate animation blending setups. See for example Unreal Engine Animation Blending.