  We analyze the asymptotic behavior of certain twisted orbital integrals
arising from the study of affine Deligne-Lusztig varieties. The main tools
include the Base Change Fundamental Lemma and $q$-analogues of the Kostant
partition functions. As an application we prove a conjecture of Miaofen Chen
and Xinwen Zhu, relating the set of irreducible components of an affine
Deligne-Lusztig variety modulo the action of the $\sigma$-centralizer group to
the Mirkovic-Vilonen basis of a certain weight space of a representation of the
Langlands dual group.
