  Let $X$ be a building, identified with its Davis realisation. In this paper,
we provide for each $x\in X$ and each $\eta$ in the visual boundary $\partial
X$ of $X$ a description of the geodesic ray bundle $Geo(x,\eta)$, namely, of
the reunion of all combinatorial geodesic rays (corresponding to infinite
minimal galleries in the chamber graph of $X$) starting from $x$ and pointing
towards $\eta$. When $X$ is locally finite and hyperbolic, we show that the
symmetric difference between $Geo(x,\eta)$ and $Geo(y,\eta)$ is always finite,
for $x,y\in X$ and $\eta\in\partial X$. This gives a positive answer to a
question of Huang, Sabok and Shinko in the setting of buildings. Combining
their results with a construction of Bourdon, we obtain examples of hyperbolic
groups $G$ with Kazhdan's property (T) such that the $G$-action on its Gromov
boundary is hyperfinite.
