The limiting reagent concept doesn't have a whole lot to do with equilibrium, per se, because it usually applies under situations where reactions are designed to go to "completion". This doesn't mean equilibrium doesn't exist - it does - but rather the reaction is so favorable that at equilibrium almost all of the species are (in principle) the desired products rather than reactants.

E.g., in a reaction

A + B

C

An event in which A and B collide almost always gives C (due to thermodynamics, say), such that, given enough time, every A in the solution reacts with every B to form one C. In this situation, K >> 1, and we would just simplify it as

A + B

C

even though implicitly some C always converts back to A + B. Now, under this simplification, if you have a lot more A than B, eventually all of the B will be consumed and there will be A left over. B is your limiting reagent. In reality, not all of B is truly consumed because there IS an equilibrium, even if it's very far to the right.

You can think of it in terms of probability.

Assume a situation

A + B

C + D

where you start with a lot of A and B. At early times there will be a large probability, per unit time, of an A and B colliding to form a C and D. Conversely, there is not a lot of C and D in solution, so there is a small probability of the reverse reaction to form A and B. The instantaneous forward reaction rate is high and the backward rate is low. As time goes on, there is less and less A and B in solution, so the probability of a forward reaction event decreases; conversely, there is more and more C and D in solution, so the probability of a reverse reaction revent increases. At some point, the probabilities are equal, such that the amount of A + B

C + D and the amount of C + D

A + B is equivalent. This is equilibrium.

This situation doesn't change if you start with an unequal (non-stoichiometric) amount of A and B in your reaction pot, or if you continually add A as a function of time. the concept of probability still applies. If you continually add A, you may raise the probability of a forward reaction event for a while, but eventually the amount of B in solution gets so low that again there is a low probability of a collision between A and B, so further adding more A doesn't do a whole lot. You are still governed by the equilibrium constant, which incorporates the idea of probability and reaction thermochemistry into the balanced forward and backward rates. You may add so much A that it "drives" the reaction forward by maximizing the probability of a reaction event between A and B, so you may consider B to be a limiting reagent in this case, and the reaction essentially "goes to completion". Just so, you can remove your C and D as they are produced, to again change the probability of A + B going forward relative to C + D going backward. These are both strategies used in synthetic chemistry to maximize product formation.