  The approach to nonholonomic Ricci flows and geometric evolution of regular
Lagrange systems [S. Vacaru: J. Math. Phys. \textbf{49} (2008) 043504 \& Rep.
Math. Phys. \textbf{63} (2009) 95] is extended to include geometric mechanics
and gravity models on Lie algebroids. We prove that such evolution scenarios of
geometric mechanics and analogous gravity can be modelled as gradient flows
characterized by generalized Perelman functionals if an equivalent
geometrization of Lagrange mechanics [J. Kern, Arch. Math. (Basel) \textbf{25}
(1974) 438] is considered. The R. Hamilton equations on Lie algebroids
describing Lagrange-Ricci flows are derived. Finally, we show that geometric
evolution models on Lie algebroids are described by effective thermodynamical
values derived from statistical functionals on prolongation Lie algebroids.
