The machine (10) comprises: a stationary base (12); at least two working heads (18) for carrying out boring, tapping, drilling, flaring, spot-facing or rolling operations on a profile (P) along a horizontal working axis perpendicular to the longitudinal axis of the profile (P); a workpiece clamping unit (30) for clamping the profile (P) while the working heads (18) are carrying out the above-mentioned operations on the profile (P); and a workpiece handling unit comprising two or more gripping devices (42) movable in a controlled manner along the direction of the longitudinal axis of the profile (P) to longitudinally position the profile (P) in a given position relative to the workpiece clamping unit (30), each gripping device (42) comprising a support body (50) and a pair of jaws (52) movable transversely towards and away from each other to clamp or release the profile (P). Each gripping device (42) is configured so as to have at least one degree of freedom (DOF 1 , DOF 2 , DOF 3 ) with controlled compliance.