  We solve the geodesic equation in the space of K\"ahler metrics under the
setting of asymptotically locally Euclidean (ALE) K\"ahler manifolds and we
prove global $\mathcal{C}^{1,1}$ regularity of the solution. Then, we relate
the solution of the geodesic equation to the uniqueness of scalar-flat ALE
metrics. To this end, we study the asymptotic behavior of
$\varepsilon$-geodesics at spatial infinity. Under the assumption that the
Ricci curvature of a reference ALE K\"ahler metric is non-positive, convexity
of the Mabuchi $K$-energy along $\varepsilon$-geodesics. However, we will also
prove that on the line bundle $\mathcal{O}(-k)$ over
$\mathbb{C}\mathbb{P}^{n-1}$ with $n \geq 2$ and $k \neq n$, no ALE K\"ahler
metric can have non-positive (or non-negative) Ricci curvature.
