In this paper we report on adiabatic pumping in quasiperiodic stiffness-modulated beams. We show that distinct topological states populating nontrivial gaps can nucleate avoided crossings characterized by edge-to-edge transitions. Such states are inherently coupled when a smooth variation of the modulation phase is induced along a synthetic dimension, resulting in topological edge-to-edge transport stemming from distinct polarizations of the crossing states. We first present a general framework to estimate the required modulation speed for a given transition probability in time. Then this analysis tool is exploited to tailor topological pumping in a stiffness-modulated beam.