The transport of polymers across nanoscale pores underpins many biological processes, such as the ejection of bacteriophage DNA into a host cell and the transfer of genes between bacteria. The movement of polymers into and out of confinement is also the basis for a wide range of sensing technologies used for single-molecule detection and sequencing. Acquiring an accurate understanding of the translocation dynamics is an essential step in the quantitative analysis of polymer structure, including the localization of binding sites or sequences. Here, we use synthetic nanopores and nanostructured DNA molecules to directly measure the velocity profile of driven polymer translocation through synthetic nanopores. Our results reveal a two-stage behaviour in which the translocation initially slows with time before accelerating close to the end of the process. We also find distinct local velocity correlations as the DNA polymer chain passes through the nanopore. Brownian dynamics simulations show that the two-stage behaviour is associated with tension propagation, with correlations arising from the random-walk conformation in which the DNA begins.