Selective excitation of a diffusive system's transmission eigenchannels enables manipulation of its internal energy distribution. The fluctuations and correlations of the eigenchannel's spatial profiles, however, remain unexplored so far. Here we show that the depth profiles of high-transmission eigenchannels exhibit low realization-to-realization fluctuations. Furthermore, our experimental and numerical studies reveal the existence of interchannel correlations, which are significant for low-transmission eigenchannels. Because high-transmission eigenchannels are robust and independent from other eigenchannels, they can reliably deliver energy deep inside turbid media.