Motivated by Krioukov et al.'s model of random hyperbolic graphs Krioukov et al. (Phys. Rev. E 82 (2010) 036106) for real-world networks, and inspired by the analysis of a dynamic model of graphs in Euclidean space by Peres et al. (Probab. Theory Related Fields 156 (2013) 273-305), we introduce a dynamic model of hyperbolic graphs in which vertices are allowed to move according to a Brownian motion maintaining the distribution of vertices in hyperbolic space invariant. For different parameters of the speed of angular and radial motion, we analyze tail bounds for detection times of a fixed target and obtain a complete picture, for very different regimes, of how and when the target is detected: as a function of the time passed, we characterize the subset of the hyperbolic space where particles typically detecting the target are initially located. Our analysis shows that our dynamic model exhibits a phase transition as a function of the relation of angular and radial speed. We overcome several substantial technical difficulties not present in Euclidean space, and provide a complete picture on tail bounds. On the way, moreover, we obtain results for a class of one-dimensional continuous processes with drift and reflecting barrier, concerning the time they spend within a certain interval. We also derive improved bounds for the tail of independent sums of Pareto random variables.