  We study the equidistribution of orbits of the form $b_1^{a_1(n)}...
b_k^{a_k(n)}\Gamma$ in a nilmanifold $X$, where the sequences $a_i(n)$ arise
from smooth functions of polynomial growth belonging to a Hardy field. We show
that under certain assumptions on the growth rates of the functions
$a_1,...,a_k$, these orbits are uniformly distributed on some subnilmanifold of
the space $X$. As an application of these results and in combination with the
Host-Kra structure theorem for measure preserving systems, as well as some
recent seminorm estimates of the author for ergodic averages concerning Hardy
field functions, we deduce a norm convergence result for multiple ergodic
averages. Our method mainly relies on an equidistribution result of Green-Tao
on finite polynomial orbits of a nilmanifold.
