Given a smooth manifold $M$ and a smooth Lagrangian $\mathcal{L}(x,\dot{x})$ on $M$, the curves which make stationary the corresponding action are those which solve the Euler-Lagrange equations.
If a non-holonomic constraint is added $F(\dot{x})=0$ for all times, what then are the EL equations?
Although I don't disagree with Nawaf Bou-Rabee's answer, I want to nuance it a bit, and this is too long for a comment.
Strictly speaking there are no "correct" equations. The Euler--Lagrange equations correspond to a vector field $X_\textrm{EL}$ on $TM$. The constraints define a subbundle (or a distribution) $\mathcal{D} \subset TM$ to which the dynamics should be restricted. However, in general, the Euler--Lagrange vector field $X_\textrm{EL}$ is not tangent to $\mathcal{D}$, so does not leave it invariant. Thus one needs to project in some way $X_\textrm{EL}|_\mathcal{D}$ onto $T\mathcal{D}$. There are in principle many ways to do this and d'Alembert's principle is one of these; for example, another way is the vakonomic principle, which means enforcing the constraints before doing variations.
The question which principle is correct then really depends on what the resulting equations are supposed to model (this was also pointed out by Kozlov in "On the realization of constraints in dynamics", Prikl. Mat. Mekh. 56-4 (1992). For typical mechanical no-slip constraints, indeed, d'Alembert's principle seems to be the (most) correct one, see Lewis and Murray "Variational principles for constrained systems: theory and experiment", Internat. J. Non-Linear Mech. 30-6 (1995).
The correct variational principle and equations for nonholonomic mechanical systems are given by the Lagrange–d’Alembert principle, where one imposes the nonholonomic constraint after taking variations. For a thorough discussion, check out Chapter 5 of
A.M. Bloch [2003]. Nonholonomic Mechanics and Control. Second Edition, 2015. Interdisciplinary Applied Mathematics, Springer.
