The effect of actively participating bounding surfaces on the free molecule flow through a slit or an annular orifice situated in a wall separating two regions of different pressure is analyzed. The flow through the slit or orifice depends on the distributions of the flux of mass leaving the bounding surfaces. These distributions are found by formulating and solving pairs of integral equations. In the case of the slit, the integral equations are formulated by employing kinetic theory methods, while for the annular orifice it was found advantageous to use the techniques of radiative transfer. In addition to exact solutions, closed-form solutions based on an approximate model are derived. Results are presented for a wide range of values of the relevant geometrical parameters.