© 2015 Elsevier Inc. We consider discrete Schrödinger operators with Sturmian potentials and study the transport exponents associated with them. Under suitable assumptions on the frequency, we establish upper and lower bounds for the upper transport exponents. As an application of these bounds, we identify the large coupling asymptotics of the upper transport exponents for frequencies of constant type. We also bound the large coupling asymptotics uniformly from above for Lebesgue-typical frequency. A particular consequence of these results is that for most frequencies of constant type, transport is faster than for Lebesgue almost every frequency. We also show quasi-ballistic transport for all coupling constants, generic frequencies, and suitable phases.