Given a fibration (or a fiber bundle or whatever might make my question true) $f\colon E\rightarrow B$ and basepoints $e$ in $E$ and $b$ in $B$, such that $f$ preserves them.
By the homotopy lifting property, I can lift every path $\gamma\colon I\rightarrow B$ starting from $b$ to a path in $E$ starting from $e$. Does this work in families, i.e. can I lift continuously varying paths in $B$ to continuously varying paths in $E$?
Let me make this into a precise question:
The restriction map $$Maps_*(I,B)\rightarrow Maps_*(I,E)$$ is surjective, where the path spaces consists of path starting at $b$ or $e$ respectively. Hence, this map has a set-theoretical section $$s\colon Maps_*(I,E)\rightarrow Maps_*(I,B)$$ and I want to know whether I can choose this section continuously and if not, how strong this assumption is, i.e. whether it will be fulfilled in most cases.