A hydrostatic transmission comprising a variable capacity hydraulic pump (45) and a hydraulic motor (46) fluidly connected with each other through a pair of fluid passages (5a,5b) is provided for driving an axle. For an internal damping system, a pair of pistons (64L,64U) having orifices (80L,80U) are connected with the respective fluid passages (5e,5f) and are pressed against a shutting member interlocking with capacity changing means (11,39) for the hydraulic pump (45), which is biased (31) to a neutral position. The shutting means is provided with one or more grooves (39i) connected with fluid sump in an apparatus housing. While the capacity changing means (11,39) returns to the neutral position or at the moment that it leaves the neutral position, at least one of the orifices (80L,80U) of the pistons (64L,64U), which is particularly connected to the higher-pressured fluid passage (5a,5b), is connected with the fluid sump through the groove (39i), so that the vehicle with the apparatus is prevented from a dynamic braking shock in deceleration or a shock on starting.