  Let $X$ be an abstract orientable not necessarily compact CR manifold of
dimension $2n+1$, $n\geq1$, and let $L^k$ be the $k$-th tensor power of a CR
complex line bundle $L$ over $X$. Suppose that condition $Y(q)$ holds at each
point of $X$, we establish asymptotics of the heat kernel of Kohn Laplacian
with values in $L^k$. As an application, we give a heat kernel proof of Morse
inequalities on compact CR manifolds. When $X$ admits a transversal CR $\mathbb
R$-action, we also establish asymptotics of the $\mathbb R$-equivariant heat
kernel of Kohn Laplacian with values in $L^k$. As an application, we get
$\mathbb R$-equivariant Morse inequalities on compact CR manifolds with
transversal CR $\mathbb R$-action.
