Using analytical techniques and Langevin dynamics simulations, we investigate the dynamics of polymer translocation into a narrow channel of width R embedded in two dimensions, driven by a force proportional to the number of monomers in the channel. Such a setup mimics typical experimental situations in nano/microfluidics. During the translocation process if the monomers in the channel can sufficiently quickly assume steady state motion, we observe the scaling τ ∼ N∕F of the translocation time τ with the driving force F per bead and the number N of monomers per chain. With smaller channel width R, steady state motion cannot be achieved, effecting a nonuniversal dependence of τ on N and F. From the simulations we also deduce the waiting time distributions under various conditions for the single segment passage through the channel entrance. For different chain lengths but the same driving force, the curves of the waiting time as a function of the translocation coordinate s feature a maximum located at identical s(max), while with increasing the driving force or the channel width the value of s(max) decreases.