The vehicle has two groups of displacement members (1, 2, 3, 4), one of which groups is steerable. The apparatus comprises a closed transmission circuit which includes a pump (50) and hydraulic motors (10, 20, 30, 40). A first group of motors that serves to drive two displacement members on the same axle comprises at least two double motors (30, 40), each having two operating cubic capacities forming two elementary motors (31, 32; 41, 42). The first elementary motor (31, 41) of each of the double motors is connected to the pump (50) in parallel with a motor (10, 20) of the other group, while the second elementary motor (32, 42) is connected in series with said motor of the other group. There thus exist first and second series ducts (C32, C42) between the motors. The series ducts are connected to a low-pressure circuit via means (66) for limiting the pressure in the series ducts (C32, C42), and the apparatus further comprises means (R58, R60, R62) for limiting the flow rate of fluid flowing between the series ducts and the low-pressure circuit.