  The decays of the $Z_b(10610)$ and the $Z_b(10650)$ to $\Upsilon(nS)\pi$,
$h_b(mP)\pi$ and $\chi_{bJ}(mP)\gamma$ ($n=1,2,3$, $m=1,2$ and $J=0,1,2$) are
investigated within a nonrelativistic effective field theory. It is argued
that, while the decays to $\Upsilon(nS) \pi$ suffer from potentially large
higher order corrections, the $P$-wave transitions of the $Z_b$ states offer
the best possibility to confirm the nature of the $Z_b$ states as molecular
states and to further study their properties. We give nontrivial and
parameter-free predictions for the ratios of various partial widths for final
states with $h_b(mP)\pi$ and $\chi_{bJ}(mP)\gamma$. In addition, the branching
fractions for the neutral $Z_b$-states to $\chi_{bJ}\gamma$ are predicted to be
of order $10^{-4}$--$10^{-3}$. This provides a fine test of the molecular
nature in future high-luminosity experiments.
