We show that optically active quantum dots (QDs) embedded in MBE-grown GaAs/AlGaAs core-shell nanowires (NWs) are coupled to the NW mechanical motion. Oscillations of the NW modulate the QD emission energy in a broad range exceeding 14 meV. Furthermore, this opto-mechanical interaction enables the dynamical tuning of two neighboring QDs into resonance, possibly allowing for emitter-emitter coupling. Both the QDs and the coupling mechanism, i.e. material strain, are intrinsic to the NW structure and do not depend on any functionalization or external field. Such systems open up the prospect of using QDs to probe and control the mechanical state of a NW, or conversely of making a quantum nondemolition readout of a QD state through a position measurement.