Tian's classical theorem shows that, for a polarized complex projective manifold, the Bergman kernel grows asymptotically like a monic polynomial of degree equal to the dimension. I will present a weighted extension of this result that includes, for instance, equivariant and divisor-adapted Bergman kernels. In contrast to the classical case, the leading coefficient now reflects the speed of a geodesic ray in Mabuchi geometry associated with the weight. In the equivariant case, this speed coincides with the Hamiltonian of the induced circle action, while in the divisor case, it arises via a resolution of the Monge-Ampère equation on the deformation to the normal cone. Finally, I will discuss how this equidistribution result connects to the theory of canonical metrics, and in particular, its implications for the Wess-Zumino-Witten equation.