  We study first-order as well as infinitary logics extended with quantifiers
closed upwards under embeddings. In particular, we show that if a chain of
quasi-homogeneous structures is sufficiently long then a given formula of such
a logic is eventually equivalent to a quantifier-free formula in that chain. We
use this fact to produce a number of undefinability results for logics with
embedding-closed quantifiers. In the final section we introduce an
Ehrenfeucht-Fra\"iss\'e game that characterizes the $L$-equivalence between
structures, where $L$ is the infinitary logic $L_{\infty \omega}$ extended with
all embedding-closed quantifiers. In conclusion, we provide an application of
this game illustrating its use.
