We use a combination of high-resolution very long baseline interferometry (VLBI) radio and multiwavelength flux density and polarization observations to constrain the physics of the dissipation mechanism powering the broadband flares in 3C 279 during an episode of extreme flaring activity in 2013-2014. Six bright flares superimposed on a long-term outburst are detected at γ-ray energies. Four of the flares have optical and radio counterparts. The two modes of flaring activity (faster flares sitting on top of a long-term outburst) present at radio, optical, and γ-ray frequencies are missing in X-rays. X-ray counterparts are only observed for two flares. The first three flares are accompanied by ejection of a new VLBI component (NC2), suggesting the 43 GHz VLBI core as the site of energy dissipation. Another new component, NC3, is ejected after the last three flares, which suggests that the emission is produced upstream from the core (closer to the black hole). The study therefore indicates multiple sites of energy dissipation in the source. An anticorrelation is detected between the optical percentage polarization (PP) and optical/γ-ray flux variations, while the PP has a positive correlation with optical/γ-ray spectral indices. Given that the mean polarization is inversely proportional to the number of cells in the emission region, the PP versus optical/γ-ray anticorrelation could be due to more active cells during the outburst than at other times. In addition to the turbulent component, our analysis suggests the presence of a combined turbulent and ordered magnetic field, with the ordered component transverse to the jet axis.