A hydraulically driven multicylinder diaphragm pumping machine for pumping difficult-to-pump materials comprises pump cylinders each having at one end (20) an inlet/outlet (21) for fluid material to be pumped supplied under adjustable pneumatic pressure, and at the other end (30) an inlet and outlet (31,32) for hydraulic oil. A separator (40) movable to-and-fro inside the pump cylinder (10) is connected to the fluid-material end of the cylinder by a bellows-like flexible diaphragm (45) and to the fluid material end (30) by another bellows-like diaphragm (46) leaving an outer annular space (49)that contains hydraulic fluid. The total volume of pumping hydraulic oil in the cylinders (10.1-10.12) is maintained constant and equal to 1/2 the total displacement volume of the cylinders by a device (65) that compensates for thermal expansion and controls the necessary return flow of hydraulic oil for driving the pump. The separators (40) move with an intake stroke at constant speed for all cylinders, and with pumping strokes function of the hydraulic oil delivery, so a lesser number of separators (40) effect an intake stroke while a greater number of separators (40) effect a discharge stroke.