Starting from Langevin equations, we derive Fokker - Planck-like equations (FPLEs) for the joint distribution of displacements and velocities, p(x,v,t), for a particle in a Gaussian random force field, firstly for the inertial process (i.e. in the absence of a frictional force) with a time correlated force, and secondly, for the Brownian motion with a white-noise force. From two different forms of the Langevin equation as coupled or decoupled first-order equations, we obtain two different forms of FPLEs for each one of these processes. In the inertial case one of the FPLEs reduces to an equation studied earlier by the author, while the other coincides with the equation obtained recently by Drory from an involved time discretization. In the Brownian motion case one of the FPLEs coincides with the free-particle Kramers equation obtained from the Fokker - Planck formalism for Markov processes. For each one of these processes the exactly determined initial value solutions of the two FPLEs are found to coincide. It follows, in particular, that the Markovian character of p(x,v,t) for the Brownian motion is respected, regardless of which FPLE is used for defining it. Furthermore, for each process the two FPLEs lead to the same diffusion-like equation for the marginal distribution of displacements. The latter have been used elsewhere for studying first passage times, as well as survival probabilities in the presence of traps.