Compatibility of canonical $\mathrm{\ell }$-adic local systems on Shimura varieties

For a Shimura variety $(G,X)$ in the superrigid regime and neat level subgroup $K_0$, we show that the canonical family of $\mathrm{\ell }$-adic representations associated to a number field point $y\in {\mathrm{S}\mathrm{h}}_{K_0}(G,X)(F)$, $${\{{\rho }_{y,\mathrm{\ell }}\text{\hspace{0.17em}⁣}:\mathrm{G}\mathrm{a}\mathrm{l}(\bar{\mathbb{Q}}\mathrm{/}F)\to G^{\mathrm{a}\mathrm{d}}({\mathbb{Q}}_{\mathrm{\ell }})\}}_{\mathrm{\ell }},$$ form a compatible system of $G^{\mathrm{a}\mathrm{d}}({\mathbb{Q}}_{\mathrm{\ell }})$-representations: there is an integer $N(y)$ such that for all $\mathrm{\ell }$, ${\rho }_{y,\mathrm{\ell }}$ is unramified away from $N(y)\mathrm{\ell }$, and for all $\mathrm{\ell }\mathrm{\ne }{\mathrm{\ell }}^{\mathrm{\prime }}$ and $v\nmid N(y)\mathrm{\ell }{\mathrm{\ell }}^{\mathrm{\prime }}$, the semisimple parts of the conjugacy classes of ${\rho }_{y,\mathrm{\ell }}({\mathrm{F}\mathrm{r}\mathrm{o}\mathrm{b}}_v)$ and ${\rho }_{y,{\mathrm{\ell }}^{\mathrm{\prime }}}({\mathrm{F}\mathrm{r}\mathrm{o}\mathrm{b}}_v)$ are ($\mathbb{Q}$-rational and) equal. We deduce this from a stronger compatibility result for the canonical $G({\mathbb{Q}}_{\mathrm{\ell }})$-valued local systems on connected Shimura varieties inside ${\mathrm{S}\mathrm{h}}_{K_0}(G,X)$. Our theorems apply in particular to Shimura varieties of non-abelian type and represent the first such independence-of-$\mathrm{\ell }$ results in non-abelian type.