This paper proposes and validates a numerical method based on the unconditionally stable dual-finite volume (DFV) scheme for Kolmogorov's forward equations (KFEs) in 1-D unbounded domains, which can be optionally equipped with a mass-conservative moving mesh partial differential equation (MMPDE) method. A KFE is a conservative and linear parabolic partial differential equation (PDE) governing spatio-temporal evolution of a probability density function (PDF) of a continuous time stochastic process. A variable transformation method is proposed for effectively solving the KFEs in 1-D bounded domains. Application of the DFV scheme to a series of test cases demonstrates its satisfactory computational accuracy, robustness, and versatility for both steady and unsteady problems. Impacts of modulating a parameter in the variable transformation method on computational performance of the DFV scheme are then numerically assessed. Advantages and disadvantages of using the MMPDE method are also investigated.