An analysis of the accessibility properties of exact stationary solutions of the resistive magnetohydrodynamics equations is presented. The issue is raised of whether such solutions, which describe a particular type of magnetic reconnection, can be obtained as the final state of a time-dependent evolution of the magnetic- and velocity-field disturbances superimposed onto background fields possessing a null point. Numerical experiments indicate that, whereas for the solutions of inflow type with super-Alfv́nic background flow the imposed boundary conditions are sufficient to determine the accessed steady-state solution, in the case of outflow solutions with sub-Alfv́nic background flow the final steady-state turns out to depend on the initial disturbance as well. It is found that the directions of the characteristic curves of the evolution equations and an implicit condition on the vorticity at the stagnation point are responsible for this behavior. Implications for the maximum current density attained in the final state are discussed. An argument to explain the appearance of damped oscillations in the evolution of the disturbance fields for the outflow case, as opposed to the flux pileup characterizing the inflow case, is also provided.