A new, wide class of relativistic stochastic processes is introduced. All relativistic processes considered so far in the literature (the Relativistic Ornstein-Uhlenbeck Process as well as the Franchi-Le Jan and the Dunkel-Hänggi processes) are members of this class. The stochastic equations of motion and the associated forward Kolmogorov equations are obtained for each process in the class. The corresponding manifestly covariant transport equation is also obtained. In particular, the manifestly covariant equations for the Franchi-Le Jan and the Dunkel-Hänggi processes are derived here for the first time. Finally, the manifestly covariant approach is used to prove a new H-theorem for all processes in the class.