Given a space $X$, what conditions on $X$ can you give to ensure that the diagonal map $X\to X\times X$ is a Hurewicz cofibration? (I am happy to assume that $X$ is compactly generated weak Hausdorff, or even just Hausdorf.)

More generally, given a map $f:X\to Y$, when is $Y\to Y\times_X Y$ a Hurewicz cofibration?