A device (10) for supporting and oscillating continuous casting moulds in continuous casting plants comprises at least one support (30) suitable to support a continuous casting mould (40), said support (30) comprising a fixed assembly (31) restrained to a frame (20) of the device (10) and a movable assembly (32) that is slidably restrained to said fixed assembly (31) in a vertical direction (A) and connected to a servomechanism (38) suitable to move it in a reciprocating manner relative to the fixed assembly (31) along said axial direction (A), said movable assembly (32) comprising a plurality of channels (50, 60) suitable to allow a flow of a cooling fluid to and from a cooling circuit of said mould (40), said channels (50, 60) being supplied by supply pipes arranged along the vertical direction (A). The device(10) further comprises at least one connecting pipe (70) suitable to allow to connect a supply pipe, said connecting pipe (70) having a T shape and comprising a first duct (71) rigidly connected to the movable assembly (32) in a horizontal direction (B), as well as a second and a third duct (72, 73) extending from said first duct (71) in opposite ways along the vertical direction (A), said second and third ducts (72, 73) being respectively connected to first and second end portions (80, 81) of the fixed assembly (31) through further axially deformable ducts (100, 101) and being respectively a blind duct (72) and a flow-through duct (73) suitable to allow the cooling fluid to flow towards the first and the second ducts (71, 72). The second and third ducts (72, 73), and preferably also the first duct (71), of the at least one connecting pipe (70) have the same diameter of the supply pipes.