  We study two dispersive regimes in the dynamics of $N$ two-level atoms
interacting with a bosonic mode for long interaction times. Firstly, we analyze
the dispersive multiqubit quantum Rabi model for the regime in which the qubit
frequencies are equal and smaller than the mode frequency, and for values of
the coupling strength similar or larger than the mode frequency, namely, the
deep strong coupling regime. Secondly, we address an interaction that is
dependent on the photon number, where the coupling strength is comparable to
the geometric mean of the qubit and mode frequencies. We show that the
associated dynamics is analytically tractable and provide useful frameworks
with which to analyze the system behavior. In the deep strong coupling regime,
we unveil the structure of unexpected resonances for specific values of the
coupling, present for $N\ge2$, and in the photon-number-dependent regime we
demonstrate that all the nontrivial dynamical behavior occurs in the atomic
degrees of freedom for a given Fock state. We verify these assertions with
numerical simulations of the qubit population and photon-statistic dynamics.
