We study the long-time behavior of the $κ$-Markov local-field equation ($κ$-MLFE), which is a conditional McKean-Vlasov equation associated with interacting diffusions on the $κ$-regular tree. Under suitable assumptions on the coefficients, we prove well-posedness of the $κ$-MLFE. We also establish an H-theorem by identifying an energy functional, referred to as the sparse free energy, whose derivative along the measure flow of the $κ$-MLFE is given by a nonnegative functional that can be viewed as a modified Fisher information. Moreover, we show that the zeros of the latter functional coincide with the set of stationary distributions of the $κ$-MLFE and are also marginals of splitting Gibbs measures on the $κ$-regular tree. Furthermore, we show that for a natural class of initial conditions, the corresponding measure flow converges to one of the stationary distributions, thus demonstrating that the sparse free energy acts as a global Lyapunov function. Under mild additional conditions, in the case $κ= 2$ we prove that the sparse free energy arises naturally as the renormalized limit of certain relative entropies. We exploit this characterization to prove a modified logarithmic Sobolev inequality and establish an exponential rate of convergence of the $2$-MLFE measure flow to its unique stationary distribution.