If $p: E\to B$ is a fibration, is the map $q:M_p \to B$ from the mapping cylinder of $p$ also a fibration?

I know that it is if $p$ is trivial, or locally trivial; and I know (from Strøm's "The Homotopy Category is a Homotopy Category") that the topology of $M_p$ can be slightly modified to make $q$ a fibration.

I wonder

1. whether the modification is necessary: does anyone know of a fibration $p$ such that $q$ is not a fibration, and

2. whether the modification is necessary if we assume the spaces are compactly generated (Strøm works with all spaces).

EDIT: There is a natural injective map from mapping cylinder $M_f$ of the map $f: X\to Y$ to the join $X * Y$; the topology on $M_f$ is modified so that it coincides with the subspace topology.