Summary: We investigate the dynamics of a nonlocal model describing spontaneous cell polarization. It consists of a drift-diffusion equation set in the half-space, with the coupling involving the trace value on the boundary. We characterize the following behaviors in the one-dimensional case: solutions are global if the mass is below the critical mass and they blow up in finite time above the critical mass. The higher-dimensional case is also discussed. The results are reminiscent of the classical Keller-Segel system, but critical spaces are different (\(L^N\) instead of \(L^{N/2}\) due to the coupling on the boundary). In addition, in the one-dimensional case we prove quantitative convergence results using relative entropy techniques. This work is complemented with a more realistic model that takes into account dynamical exchange of molecular content at the boundary. In the one-dimensional case we prove that blow-up is prevented. Furthermore, density converges toward a nontrivial stationary configuration.