There is no direct equivalent, as you say existing plants either move very slowly and in a permanent way (by growing towards the light) or have a limited movement such as in a venus fly-trap.
Carnivorous plants use a number of different mechanisms to move, but they could mostly be described as cells changing size and shape in order to cause the plant to curl in a certain way.
It seems that the most likely way a plant would evolve faster and more powerful reason might be hydraulics. The existing circulatory system for fluids could be strengthened and pressurized, that could then cause limbs to move in a controlled and powerful fashion.
The plants could have flexible joints and a rigid wooden frame either internally or externally. Hydraulics could then power the movements of the frame. The main issue would be containing and generating the pressures needed for effective hydraulics and gathering the energy needed to move.
The main problem for your moving plants isn't the actual mechanics of the movement itself. Its the fact that movement requires a lot of energy, far more than can be generated easily by sitting in the sun. For the movement to achieve things the plant needs a sensory system to detect the need to move, and a goal to move for.
You should also consider roots and similar. It's much easier to have a mostly stationary plant capable of occasional bursts of movement to capture a passing prey animal than it is to imagine one that can uproot itself and move around.