It comprises an infusion chamber (2) for housing a dose of product and means for supplying a liquid under pressure to said infusion chamber, said supply means comprising a hydraulic accumulator device (4a, 4b) susceptible of storing under pressure a liquid from liquid injection means (8, 8a, 8b, 9, 25) of said machine (1, 1′), characterised in that said liquid injection means (8, 8a, 8b, 9, 25) and said hydraulic accumulator device (4a, 4b) are configured and disposed interconnected to the infusion chamber (2), so that a first fraction of injected liquid is susceptible of being stored under pressure in a cavity (5) of said accumulator device (4a, 4b), while a second fraction of the injected liquid is simultaneously distributed or supplied to the exterior through the infusion chamber (2) of the machine (1, 1′), allowing said accumulator device (4a, 4b) to store a fraction of liquid configured to ensure the continuous supply of a certain flow (Qc) during a predetermined beverage preparation time or distribution time (te). The process comprises the steps of retracting the distribution cylinder (4a) piston (4b) or an element of said accumulator device to its active position through the injection of a liquid from an independent storage reservoir (9), and simultaneously supplying the liquid injected under pressure.