Let $X$ be a non-empty set and let $A$ be an algebra of bounded functions on $X$ containing the constant functions and separating the points in $X$. Given a distance $\rho$ on $X$, we define $\rho_\ell$ to be the length distance generated by $\rho$ and $\rho_A$ as the function $$\rho_A(x,y):= \sup \left \{ |f(x)-f(y)| : f \in A,\, Lip(f,X,\rho) \le 1 \right \}, \quad x,y \in X,$$ where $Lip(f,X,\rho)$ is the Lipschitz constant of $f$ on $X$ w.r.t. $\rho$. It is not difficult to show that, if functions in $A$ are $\rho$-Lipschitz, then $\rho_A$ is a distance on $X$. In this case, it is also clear that $\rho_A \le \rho \le \rho_\ell$.
What happens when we apply both operations? Namely, suppose that $d$ is a distance on $X$ and that functions in $A$ are $d$-Lipschitz, then what is the relation between $(d_A)_\ell$ and $(d_\ell)_A$? Under which conditions on $A,X$ and $d$ we have that $(d_\ell)_A \le (d_A)_\ell$? For example this is the case if $A$ coincides with the whole set of $d$-Lipschitz and bounded functions (since $d_A=d$ in this case), but I am wondering if this is true in general.