The invention discloses a construction method of girder transport on a under girder and full-span erection for a large-tonnage asymmetric framing precast box girder. A girder transport device comprises a bridge erecting machine and a girder transport vehicle, and the bridge erecting machine is of a three-leg structure and used for full-span erection of the left-right double-framing large-tonnage precast box girder, and can turn around on a bridge through exchanging a front leg and a rear leg so that reverse girder erection can be conducted; the girder transport vehicle is of a single vehicle structure, and can enter the tail part of the bridge erecting machine bi-directionally and directly for adding a girder. The construction method comprises the steps that one girder transport vehicle carrying the girder crosses a web arranged at the internal side of the erected left-right double-framing large-tonnage precast box girder to transport the large-tonnage asymmetric framing precast box girder; the bridge erecting machine moves longitudinally and passes through a hole; the bridge erecting machine conducts full-span erection on the left-right double-framing large-tonnage precast box girder, and so on. By means of the construction method, girder transport on a under girder of the large-tonnage asymmetric framing precast box girder through one girder transport vehicle and full-span erection on the under girder of the left-right double-framing large-tonnage precast box girder through the bridge erecting machine are achieved, and moreover the construction method has the advantages of being safe in operation and high in efficiency.