The motivation for this is demonstrating that the proof in Simply connected manifold with nonpositive curvature has no more than one geodesic between points is correct (or learning that it is false).
My argument for the statement being true is that a homotopy gives an $f:[0,1]\times[0,L]\to M$, with $f_0,f_1$ smooth. Since every point has a convex neighborhood and $[0,1]\times[0,L]$ is compact, we can take some positive $\epsilon_0$ such that $f(s+\epsilon,t)$ stays within a convex neighborhood of some maximal radius $r_0$ of $f(s,t)$ when $\epsilon<\epsilon_0$.
Then we can take $s$ in increments of $\frac{\epsilon_0}{2}$ and take a homotopy of $f(k*\frac{\epsilon_0}{2},t)$ to some $\tilde{f}$ smooth in $t$ by moving to points within $\frac{r_0}{3}$, thus staying within the convex neighborhoods of the original points. Thus we can bridge the gap between increments of $s$ by the exponential map of something smooth in $s$ from $\tilde{f}(k*\frac{\epsilon_0}{2},t)$ to $\tilde{f}((k+1)*\frac{\epsilon_0}{2},t)$, giving a smooth variation from $f_0$ to $f_1$. The vectors can be chosen to match on both sides of each increment, so that the smoothness continues across each increment of $s$.