  We study $\omega$-weak equivalences between weak $\omega$-categories in the
sense of Batanin-Leinster. Our $\omega$-weak equivalences are strict
$\omega$-functors satisfying essential surjectivity at every dimension, and
when restricted to those between strict $\omega$-categories, they coincide with
the weak equivalences in the model category of strict $\omega$-categories
defined by Lafont, M\'etayer, and Worytkiewicz. We show that the class of
$\omega$-weak equivalences has the 2-out-of-3 property. We also consider a
generalisation of $\omega$-weak equivalences, defined as weak $\omega$-functors
(in the sense of Garner) satisfying essential surjectivity, and show that this
class also has the 2-out-of-3 property.
