  In this work we study a scenario where dynamics is restricted to a single
excitation manifold, for particular physical observables with support in the
manifold, which we label a `correlated quench'. We ask how such dynamics may in
general differ from predictions of the eigenstate thermalization hypothesis
(ETH). We show that if thermalization occurs, it will not fulfil other key
predictions of the ETH; instead following differing generic behaviours. We show
this by analysing long-time fluctuations, two-point correlation functions, and
the out-of-time-ordered correlator; analytically detailing deviation from ETH
predictions. We derive instead an ETH-like relation, with non-random
off-diagonals, for matrix elements of observables, with correlations that alter
long-time behaviour and constrain dynamics. Further, we analytically compute
the time-dependence of the decay to equilibrium, showing that it is
proportional to the survival probability of the initial state. We finally note
that the conditions studied are common in many physical scenarios, such as
under the rotating-wave approximation. We show numerically that predictions are
robust to perturbations that break this approximation.
