Among the transversally elliptic operators initiated by Atiyah and Singer, Kohn’s lb operator on CR manifolds with $S^{1}$ action is a natural one of geometric significance for complex analysts. Our first main result establishes an asymptotic expansion for the heat kernel of such an operator with values in its Fourier components, which involves a contribution in terms of a distance function from lower dimensional strata of the $S^{1}$-action. Our second main result computes a local index density, in terms of tangential characteristic forms, on such manifolds including Sasakian manifolds of interest in String Theory, by showing that certain non-trivial contributions from strata in the heat kernel expansion will eventually cancel out by applying Getzler’s rescaling technique to off-diagonal estimates. This leads to a local result which can be thought of as a type of local index theorem on these CR manifolds. We give examples of these CR manifolds, some of which arise from Brieskorn manifolds. Moreover in some cases, we can reinterpret Kawasaki’s Hirzebruch-Riemann-Roch formula for a complex orbifold equipped with an orbifold holomorphic line bundle, as an index theorem obtained by a single integral over a smooth CR manifold. We achieve this without use of equivariant cohomology method and our method can naturally drop the contributions arising from lower dimensional strata as done in previous works.