Invariant equations of motion under Lorentz transformations

My question regards the statement that an equation of motion may be invariant under a Lorentz transformation

I just finished watching the Stanford University special relativity lectures on special relativity, taught by Leonard Susskind.

Throughout the field theory part of the lectures, he teaches us that using invariant quantities such as the proper time, we can form Lagrangians that are invariant under Lorentz transformations and thus the equations of motions derived from the Euler-Lagrange equations will also be invariant.

My question is: Does this mean that a moving observer and a stationary observer will both agree on the equations? I assume the answer is yes but I'm not sure how to get my head around it. If the moving observer measures different lengths and times than the stationary observer, I don't see how they can both agree on the trajectory of a particle or the dynamics of a field or some general system when generally they have these measurement discrepancies.

$$\ddot{\vec{x}}(t) = 0.$$