Let $(X,A)$ and $(Y,B)$ be pairs of spaces and subspaces, let $\operatorname{Map}(X,Y)$ the space of maps $f:X\to Y$ equipped with the compact-open topology and let $\operatorname{Map}(X,A;Y,B)$ be the subspace of maps $f:X\to Y$ such that $f(A)\subseteq B$. Suppose the inclusions $A\hookrightarrow X$ and $B\hookrightarrow Y$ are cofibrations, would that be enough to ensure the inclusion $\operatorname{Map}(X,A;Y,B)\hookrightarrow\operatorname{Map}(X,Y)$ is a cofibration or are other conditions needed?

In particular if $X$ and $Y$ are well-pointed is the inclusion of the based mapping space $\operatorname{Map}_*(X,Y)\hookrightarrow\operatorname{Map}(X,Y)$ a cofibration?

It seems like this should be true for reasonably nice spaces and there are similar results. I know, for example, that if $B\hookrightarrow Y$ is a closed cofibration and $X$ is compact Hausdorff then the inclusion $\operatorname{Map}(X,B)\hookrightarrow\operatorname{Map}(X,A;Y,B)$ is a cofibration. In particular this makes based mapping spaces with compact Hausdorff domain well-pointed.