We show that low-angle grain boundaries (GB) in high-temperature superconductors exhibit intermediate Abrikosov vortices with Josephson cores, whose length \(l\) along GB is smaller that the London penetration depth, but larger than the coherence length. We found an exact solution for a periodic vortex structure moving along GB in a magnetic field \(H\) and calculated the flux flow resistivity \(R_F(H)\), and the nonlinear voltage-current characteristics. The predicted \(R_F(H)\) dependence describes well our experimental data on \(7^{\circ}\) unirradiated and irradiated \(YBa_2Cu_3O_7\) bicrystals, from which the core size \(l(T)\), and the intrinsic depairing density \(J_b(T)\) on nanoscales of few GB dislocations were measured for the first time. The observed temperature dependence of \(J_b(T)=J_{b0}(1-T/T_c)^2\) indicates a significant order parameter suppression in current channels between GB dislocation cores.