  The existence of new vector-like quarks is often predicted by models of new
physics beyond the Standard Model, and the development of discovery strategies
at colliders is the object of an intense effort from the high-energy community.
Our analysis aims at identifying the constraints on and peculiar signatures of
simplified scenarios containing \textit{two} vector-like quark doublets mixing
with \textit{any} of the SM quark generations. This scenario is a necessary
ingredient of a broad class of theoretically motivated constructions. We focus
on the two charge $2/3$ states $t_{1,2}^\prime$ that, due to their peculiar
mixing patterns, feature new production and decay modes that are not searched
for at the LHC: single production of the heavier state can dominate over the
light one, while pair production via electroweak interactions overcomes the QCD
one for masses at the TeV scale.
