  The dipion transitions $\Upsilon(10860)\to\pi^+\pi^-\Upsilon(nS)$ ($n=1,2,3$)
are studied in the framework of a unitary and analytic coupled-channel
formalism previously developed for analysing experimental data on the
bottomoniumlike states $Z_b(10610)$ and $Z_b(10650)$ [Phys. Rev. D 98, 074023
(2018)] and predicting the properties of their spin partners [Phys. Rev. D 99,
094013 (2019)]. In this work we use a relatively simple but realistic version
of this approach, where the scattering and production amplitudes are
constructed employing only short-ranged interactions between the open- and
hidden-flavour channels consistent with the constraints from heavy quark spin
symmetry, for an extended analysis of the experimental line shapes. In
particular, the transitions from the $\Upsilon(10860)$ to the final states $\pi
\pi h_b(mP)$ ($m=1,2$) and $\pi B^{(*)}\bar B^* $ already studied before, are
now augmented by the $\Upsilon(10860)\to\pi^+\pi^-\Upsilon(nS)$ final states
($n=1,2,3$). This is achieved by employing dispersion theory to account for the
final state interaction of the $\pi\pi$ subsystem including its coupling to the
$K\bar K$ channel. Fits to the two-dimensional Dalitz plots for the
$\pi^+\pi^-\Upsilon$ final states were performed. Two real subtraction
constants are adjusted to achieve the best description of the Dalitz plot for
each $\Upsilon(nS)$ $(n=1,2,3)$ while all the parameters related to the
properties of the $Z_b$'s are kept fixed from the previous study. A good
overall description of the data for all
$\Upsilon(10860)\to\pi^+\pi^-\Upsilon(nS)$ channels achieved in this work
provides additional strong support for the molecular interpretation of the
$Z_b$ states.
