We compare total flux density variations in 27 γ-ray blazars with structural changes in their parsec-scale jets using multi-epoch VLBA observations at 22 and 43 GHz together with data from the Metsähovi quasar monitoring program at 22 and 37 GHz. There is a clear connection between total flux density outbursts and VLBI components emerging into the jet. For essentially every new moving VLBI component, there is a coincident total flux density flare, with evolution similar to that of the component. Furthermore, extrapolated ejection times of the new VLBI components correspond to the beginnings of associated flares. Our results suggest that it is possible to explain all the radio variations as shocks propagating down the jet. A large fraction of the shocks grow and decay within the innermost few tenths of a milliarcsecond and therefore we see them only as "core flares" in the VLBI images. However, with present data we cannot exclude the possibility that the core itself also brightens (and thus contributes to the flare) as a shock passes through it.