  Track functions describe the collective effect of the fragmentation of quarks
and gluons into charged hadrons, making them a key ingredient for jet
substructure measurements at hadron colliders, where track-based measurements
offer superior angular resolution. The first moment of the track function,
describing the average energy deposited in charged particles, is a simple and
well-studied object. However, measurements of higher-point correlations of
energy flow necessitate a characterization of fluctuations in the hadronization
process, described theoretically by higher moments of the track function. In
this paper we derive the structure of the renormalization group (RG) evolution
equations for track function moments. We show that energy conservation gives
rise to a shift symmetry that allows the evolution equations to be written in
terms of cumulants, $\kappa(N)$, and the difference between the first moment of
quark and gluon track functions, $\Delta$. The uniqueness of the first three
cumulants then fixes their all-order evolution to be DGLAP, up to corrections
involving powers of $\Delta$, that are numerically suppressed by an effective
order in the perturbative expansion for phenomenological track functions.
However, at the fourth cumulant and beyond there is non-trivial RG mixing into
products of cumulants such as $\kappa(4)$ into $\kappa(2)^2$. We analytically
compute the evolution equations up to the sixth moment at
$\mathcal{O}(\alpha_s^2)$, and study the associated RG flows. These results
allow for the study of up to six-point correlations in energy flow using
tracks, paving the way for precision jet substructure at the LHC.
