The free smooth path space on a manifold Let $M$ be a closed, smooth manifold and let $PM$ be the space of unbased piecewise smooth paths $[0,1] \to M$. Then restricting a path to its boundary gives a map
$$
PM \to M \times M .
$$
Question is this map a fiber bundle?
Andrew Stacey showed that a related map, the free smooth loop fibration $LM \to M$, is a fiber bundle (see The differential topology of loop spaces, arXiv:math/0510097). However, an inspection of his method shows that it does not immediately adapt to the situation above.   
 A: Yes.
The technical details are in Yet More Smooth Mapping Spaces and Their Smoothly Local Properties, specifically in Section 5 which establishes that smooth manifolds are smoothly locally deformable which means that there are lots of diffeomorphisms flying around.
Interestingly, although I considered subspaces I didn't consider spaces over other spaces.  Nonetheless, the same technology allows us to do so.
Let $M$ be a smooth manifold.  Section 5 of Yet More ... shows that $M$ is smoothly locally deformable.  In the discussion preceding Proposition 3.12 it is shown that this means that there is a neighbourhood $M \subseteq V \subseteq M \times M$ and a smooth map $\phi \colon \mathbb{R} \times V \to \operatorname{Diff}(M)$ with the following properties:


*

*For $v \in V$, $t \mapsto \phi_{t,v}$ is a group homomorphism $(\mathbb{R},+) \to \operatorname{Diff}(M)$.

*For $t \in \mathbb{R}$ and $v = (x,y) \in V$, $\phi_{t,v}$ is the identity outside $V_x := \{x' : (x,x') \in V\}$.

*For $v = (x,y) \in V$, $\phi_{1,v}(y) = x$.


Now let $T$ be a compact smooth space and $S \subseteq T$ a compact subset.  We assume that there is a neighbourhood $S \subseteq U \subseteq T$ with a retraction $\tau \colon U \to S$, and a bump function $\sigma \colon T \to [0,1]$ such that $\sigma(S) \subseteq \{1\}$ and $\overline{\sigma^{-1} (0,1]} \subseteq U$.
Fix a class of function that is closed under diffeomorphism and which satisfies a sheaf condition in that functions can be defined locally.
Let $\alpha \colon S \to M$ be a function.  Define $C\big((S,T),(V,M)\big)_\alpha$ to be the space of functions $\beta \colon T \to M$ with the property that $(\alpha, \beta\mid_S)$ maps $S$ into $V$.  Define $C(T,M)_\alpha$ to be the space of functions $\beta \colon T \to M$ such that $\beta\mid_S = \alpha$.  Define $C(S,V)_\alpha$ to be the space of functions $\beta \colon S \to M$ such that $(\alpha,\beta)$ maps $S$ into $V$ (I'm not sure my notation is the best here!).
We define $\Phi \colon C\big((S,T), (V,M)\big)_\alpha \to C(T,M)_\alpha \times C(S,V)_\alpha$ as follows.  The map to the second factor is simply the restriction to $S$.  The map to the first factor takes a function $\beta \colon T \to M$ to the function:
$$
t \mapsto \begin{cases}
\phi_{\sigma(t), (\alpha(\tau(t)), \beta(\tau(t)))}\big(\beta(t)\big) & t \in U \\\\
\beta(t) & t \notin U
\end{cases}
$$
The conditions on $\phi$ mean that this patches together to give a well-defined function.  The inverse of $\Phi$ takes a pair $(\beta,\gamma)$ to:
$$
t \mapsto \begin{cases}
\phi_{-\sigma(t), (\alpha(\tau(t)),\gamma(\tau(t)))}\big(\beta(t)\big) & t \in U \\\\
\beta(t) & t \notin U
\end{cases}
$$
The case in point uses piecewise-smooth functions, $T = [0,1]$ and $S = \{0,1\}$.    The conditions are easily checked.
Further Reading


*

*The differential topology of loop spaces, particularly Proposition 5.1.  This contains the germ of the idea.

*Yet More Smooth Mapping Spaces and Their Smoothly Local Properties, this contains the technical results needed.  Proposition 3.12 is quite close to what you need here.  This would establish that $LM \subseteq PM$ has a tubular neighbourhood, which says that it is a bundle on a neighbourhood of a diagonal.  Interestingly, I didn't consider fibrations of mapping spaces.  Maybe I should add another section ...

*The Smooth Structure of the Space of Piecewise-Smooth Loops about piecewise-smooth maps.
