Magnetically driven non-stationary acceleration of jets in active galactic nuclei results in the leading parts of the flow being accelerated to much higher Lorentz factors than in the case of steady-state acceleration with the same parameters. The higher Doppler-boosted parts of the flow may dominate the high-energy emission of blazar jets. We suggest that highly variable GeV and TeV emission in blazars is produced by the faster moving leading edges of highly magnetized non-stationary ejection blobs, while the radio data trace the slower-moving bulk flow. Thus, the radio and gamma-ray emission regions have different, but correlated, Doppler factors. High-energy emission is generated, typically within the optically thick core, in the outer parts of the broad-line emission region, avoiding the radiative drag on the faster parts of the flow. The radio emission should correlate with the gamma-ray emission, delayed with frequency-dependent time lag of the order of weeks to months. Model predictions compare favorably with the latest Fermi γ -ray and MOJAVE radio very long baseline interferometry results.