We consider the following dynamic Boolean model introduced by van den Berg, Meester and White (1997). At time $0$, let the nodes of the graph be a Poisson point process in $R^d$ with constant intensity and let each node move independently according to Brownian motion. At any time $t$, we put an edge between every pair of nodes if their distance is at most $r$. We study two features in this model: detection (the time until a target point--fixed or moving--is within distance $r$ from some node of the graph), coverage (the time until all points inside a finite box are detected by the graph) and percolation (the time until a given node belongs to the infinite connected component of the graph). We obtain asymptotics for these features by combining ideas from stochastic geometry, coupling and multi-scale analysis. This is joint work with Yuval Peres, Alistair Sinclair and Alexandre Stauffer.