Let $C$ be a complete category, let $I$ be a small category, let $F,G:I\to C$ be functors, and let $W,U:C\to\mathrm{Set}$ be also functors, which we view as "weights". The weighted limits are then defined, $\lim_W F$ and $\lim_U G$.
Suppose now that we have natural transformations $\alpha:F\Rightarrow G$ and $\beta:W\Rightarrow U$. Do $\alpha$ and $\beta$ induce a morphism $\lim_W F\to\lim_U G$, analogously to what happens with ordinary limits?
Let $X$ be an arbitrary object in $\mathcal{C}$. I write $\{ W, F \}$ for the limit of $F$ weighted by $W$. By definition, $$\mathcal{C} (X, \{ W, F \}) \cong [\mathcal{I}, \textbf{Set}] (W, \mathcal{C} (X, F))$$ naturally in $X$. If we have $U \Rightarrow W$ (note the direction!) and $F \Rightarrow G$ then functoriality of $W$ and $F$ on the RHS gives a map $$[\mathcal{I}, \textbf{Set}] (W, \mathcal{C} (X, F)) \longrightarrow [\mathcal{I}, \textbf{Set}] (U, \mathcal{C} (X, G))$$ natural in $X$. Applying the definition of weighted limit, this can be identified with a map $$\mathcal{C} (X, \{ W, F \}) \longrightarrow \mathcal{C} (X, \{ U, G \})$$ natural in $X$. Now apply the Yoneda lemma to obtain $\{ W, F \} \to \{ U, G \}$. In short, $\{ W, F \}$ is contravariantly functorial in $W$ and covariantly functorial in $F$.
Incidentally, it is not necessary to hypothesise completeness – you have this morphism as soon as the weighted limits in question exist.
