We report on the Josephson vortex dynamics in $Bi_2Sr_2CaCuO_{8+delta}$ natural Josephson junctions by c-axis tunneling measurements. Beside the quasiparticle branches in the current-voltage characteristics, a new set of multiple branches, referred to as Josephson-vortex-flow branches (JVFBs), are observed. The JVFBs emerge in an in-plane magnetic field above $H_0;=;{Phi}_0/{gamma}s^2$ and show highly hysteretic behavior, which can be explained in terms of the recently proposed dynamic-phase-separation model. In this work we examined the effect on the JVFBs by the presence of pancake vortices generated as the external magnetic field was applied slightly tilted from the in-plane direction. JVFBs were found to become larger and prominent with increasing pancake vortex density as the tilt angle increased, which were presumably caused by slowing down of a Josephson vortex lattice in the presence of pancake vortices.