The moving finite element method (MFEM) with polynomial approximations of any degree is applied to a variety of models described by partial differential equations (PDEs) of the type Gu(t) = Fu(xx) + H, a less than or equal to x less than or equal to b, t greater than or equal to 0, G and F are full matrices. The objective of this work is to show that the proposed formulation of MFEM is a powerful tool to compute the numerical solution of time-dependent PDEs involving steep moving fronts. A strategy to choose the penalty constants was devised in relation with the ODE solver tolerances to improve the robustness of the method. Numerical results concerning combustion model, boundary layer problem, catalytic reactor and pressurization of adsorption beds illustrate the effectiveness of our scheme.