Explore kinetic optimal transport theory through a comprehensive mathematical lecture that introduces a novel second-order kinetic discrepancy between probability measures on phase space based on Newton's equations. Learn about the groundbreaking construction developed by Jan Maas, Filippo Quattrocchi, and Giovanni Brigati that establishes a fundamental connection between optimal transport and kinetic theory. Discover the main theoretical result demonstrating a one-to-one correspondence between absolutely continuous curves of measures in the kinetic setting and solutions to Vlasov's equations. Gain insights into advanced mathematical concepts at the intersection of probability theory, optimal transport, and kinetic equations through detailed proofs and theoretical foundations presented at the Erwin Schrödinger International Institute's workshop on probabilistic mass transport.