Since you are requiring, among other things, some stability for the existence of solution of $F(x)=0$ under perturbation, it seems quite natural to think in terms of topological degree.

- Assume that there is an open bounded nbd of $x,$ $\Omega\subset\mathbb{R}^d,$ such that $x$ is the only solution of $F=0$ in $\bar\Omega.$ In particular this implies that $\deg(F,\Omega,0)$ is well defined.
- Assume further $\deg(F,\Omega,0)\ne 0.$ (several different facts may ensure this: $F$ is an odd map, or it is injective; or you can compute its degree etc).
- Assume that $F_n$ converge to $F$ uniformly on $\bar\Omega$ (several diffferent facts may ensure this too: for instance, $F_n$ converges pointwise to $F, $ and it is uniformly equicontinuous on $\bar\Omega$, etc).

Then, $\deg(F_n,\Omega,0)$ is eventually defined and different to $0$. So there exists $x_n\in\Omega$ solving $F_n(x_n)=0$ (not necessarily unique). By the compactness of $\bar\Omega$ together with the uniform convergence of $F_n$ to $F$ on $\bar\Omega$, we can conclude $x_n\to x$ as we wish.

*PS*: all maps are assumed to be continuous, of course.
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*Further remark.* Generally speaking, even in more general contexts than $\mathbb{R}^d$, any existence result for solutions of $F(x)=0$ (via degree theory, topologic and metric fixed point theorems, minimization, critical point methods,... &c) will give you a perturbation result as the one you are saying, provided you have some uniform a priori bounds for the solutions of $F_n=0,$ and some compactness. Also, if the unperturbed equation $F(x)=0$ is thought as a trivial case, and what you are mainly interested in is the perturbed equation $F_n=0,$ then bifurcation theory is what you want. Lastly, one more very elementary example, for all.

*Perturbation for the contraction principle.* Assume $T_n$ is a pointwise convergent sequence of contractions, with (uniform) Lipschitz constant $k<1.$ Then the limit map $T$ is a $k$-contraction too, and the sequence $x_n$ of the fixed points of $T_n$ converges to the fixed point of $T$. Indeed, it's immediate to check that $\|x-x_n\|\le\frac{1}{1-k}\|T(x)-T_n(x)\|.$