We introduce a novel migration process, the target process. This process is dual to the zero-range process (ZRP) in the sense that, while for the ZRP the rate of transfer of a particle only depends on the occupation of the departure site, it only depends on the occupation of the arrival site for the target process. More precisely, duality associates to a given ZRP a unique target process, and vice versa. If the dynamics is symmetric, i.e., in the absence of a bias, both processes have the same stationary-state product measure. In this work we focus our interest on the situation where the latter measure exhibits a continuous condensation transition at some finite critical density ρ_c, irrespective of the dimensionality. The novelty comes from the case of asymmetric dynamics, where the target process has a nontrivial fluctuating stationary state, whose characteristics depend on the dimensionality. In one dimension, the system remains homogeneous at any finite density. An alternating scenario prevails at high density: typical configurations consist of long alternating sequences of highly occupied and less occupied sites. The local density of the latter sites is equal to ρ_c and their occupation distribution is critical. The coherence length of these alternating structures diverges quadratically at high density. In dimension two and above, the asymmetric target process exhibits a phase transition at a threshold density ρ₀ much larger than ρ_c. The system is homogeneous at any density below ρ₀, whereas for higher densities it exhibits an extended condensate elongated along the direction of the mean current, on top of a critical background with density ρ_c.