The transmission eigenchannels of a disordered medium range from unity transmission and high internal energy density to vanishing transmission and low internal energy density. Although the transmission matrix underlies central ideas in mesoscopic electronic transport, it cannot be measured directly in electronic systems. For classical waves, however, the measurable matrix gives access to eigenchannels that control transmission and energy density within the medium. Coherence underlies transmission-eigenvalue scaling, the bimodal distribution in diffusive samples, and Anderson localization. What has remained unresolved is how interference among modal contributions is organized within individual eigenchannels throughout the sample. Here, microwave transmission-matrix measurements resolve the complex contribution of each incident mode to every transmitted mode of each eigenchannel. In simulations, we construct a flux matrix that generalizes the transmission matrix to every depth and further separate each internal modal contribution into forward- and backward-propagating components. For each eigenchannel, the directional flux in every mode throughout the sample factorizes into the squared modal amplitudes of the incident eigenchannel, coupling between waveguide modes, and modal alignment. In high-transmission eigenchannels, the contributions align increasingly constructively with depth, approaching nearly perfect alignment in the highest channel near the localization crossover. In low-transmission eigenchannels, contributions interfere destructively, producing vanishing transmission at a transmission zero. Because the contributions remain appreciable as their coherent sum approaches zero, transmission far below the noise floor of conventional transmission measurements can be determined. In our measurements, the lowest transmission eigenvalue near a transmission zero is more than nine orders of magnitude below the highest. These results provide a depth-resolved decomposition of the modal contributions and interference among them, which produces the broad range of eigenchannel transmissions.