We explore the connection between the γ-ray and radio emission in the jet of the blazar 0716+714 by using 15, 37, and 230 GHz radio and 0.1–200 GeV γ-ray light curves spanning 10.5 yr (2008–2019). We find significant positive and negative correlations between radio and γ-ray fluxes in different time ranges. The time delays between radio and γ-ray emission suggest that the observed γ-ray flares originated from multiple regions upstream of the radio core, within a few parsecs from the central engine. Using time-resolved 43 GHz Very Long Baseline Array maps we identified 14 jet components moving downstream along the jet. Their apparent speeds range from 6c to 26c, and they show notable variations in their position angles upstream from the stationary component (∼0.53 mas from the core). The brightness temperature declines as a function of distance from the core according to a power law that becomes shallower at the location of the stationary component. We also find that the periods at which significant correlations between radio and γ-ray emission occur overlap with the times when the jet was oriented to the north. Our results indicate that the passage of a propagating disturbance (or shock) through the radio core and the orientation of the jet might be responsible for the observed correlation between the radio and γ-ray variability. We present a scenario that connects the positive correlation and the unusual anticorrelation by combining the production of a flare and a dip at γ-rays by a strong moving shock at different distances from the jet apex.