Nonlinear diffusive interpenetration of a finite-β plasma and a magnetic field is considered in the framework of the one-fluid magnetohydrodynamics (MHD) in the slab geometry. If the characteristic interpenetration time is much longer than the magnetoacoustic time, the process proceeds under the force equilibrium. Then the problem can be reduced to a single second-order nonlinear diffusion equation in the Lagrangian coordinates. Two examples of the interpenetration are considered. The first presents the diffusion of a plasma into a plasma in a perpendicular magnetic field. The problem proves to be self-similar, and its solution is found. The second one concerns the diffusive expansion of an initially confined plasma slab across a perpendicular magnetic field. The problem is solved numerically and the results are compared with an asymptotic self-similar solution, found earlier in the low-β limit.