I study the absorbing-state phase transition in the one-dimensional contact process with mobile disorder. In this model the dilution sites, though permanently inactive, diffuse freely, exchanging positions with the other sites, which host a basic contact process. Even though the disorder variables are not quenched, the critical behavior is affected: the critical exponents δ and z, the ratio and the moment ratio m = ⟨ρ²⟩/ρ² take values different from those of directed percolation, and appear to vary with the vacancy diffusion rate. While the survival probability starting from a single active seed follows the usual scaling, P(t)∼t^−δ, at the critical point, the mean number of active sites and mean square spread grow more slowly than power laws. The critical creation rate increases with the vacancy density v and diverges at a value v_c<1. The scaling behavior at this point appears to be simpler than for smaller vacancy densities.