Let $(X, d)$ be a compact, complete, separable metric space, and $g_n$ a sequence of constant speed geodesic with the same endpoints, i.e. continuous maps $g_n : [0,1] \rightarrow X$ such that $$ d(g_n(t), g_n(s)) = |t-s| d(g_n(1), g_n(0)) \quad \forall t,s \in [0,1], \forall n \in \mathbb N$$ and $g_n(0) = g_{n+1}(0) , g_n(1) = g_{n+1}(1)$ for every $n \in \mathbb N$.
Can I extract from $g_n$ a subsequence which converges to a constant speed geodesic? (the space of geodesics on $X$ is a compact, complete and separable metric space with distance $$\delta(f,g) = \sup_{x \in [0,1]} d(f(x),g(x)).)$$
I am not completely sure of the answer, but I think you can generalize the Arzela Ascoli theorem to the family of function $C([0,1] , X)$. The condition that all $g_n$ are geodesic implies that the family $\{g_n\}$ are equicontinuous ($X$ is compact, thus has finite diameter). Thus there is a continuous $g : [0,1]\to X$ such that a subsequence of $\{g_n\}$ converges to $g$ uniformly. This again forces $g$ to be a geodesic.