We have proposed the "railway-switch" model to describe the superconducting transport current in (Bi,Pb)2Sr2Ca2Cu3O10 silver-sheathed tapes. The model assumes that in the textured polycrystalline filament the frequent small-angle c-axis tilt grain boundaries (‘‘railway switches'') constitute strong links for the supercurrent. With the objective to identify the mechanisms that limit the critical-current density in the tapes we present measurements of the transport critical current normal to the tape plane and of the current-transfer length along the tape axis. From I-V curves we obtain the longitudinal critical-current distribution and compare it to the thickness variation of the filament. The experiments have been performed on monofilamentary powder-in-tube samples prepared in long lengths by rolling as the only tape-forming process. For all investigated samples the critical-current density at T=77 K in self-field was in the range jc=20–30 kA/cm2. We conclude that the dominant limitation for the transport critical current in the tapes is imposed by the low intragrain critical-current density jcc along the c axis (as compared to the in-plane critical-current density jabc) and by the even lower critical-current density jtc across twist boundaries or intergrowths. Possibilities for an improvement of the performance of the (Bi,Pb)2Sr2Ca2Cu3O10 silver-sheathed tapes are discussed.