After a quench in a quantum many-body system, expectation values tend to relax towards long-time averages. However, in any finite-size system, temporal fluctuations remain. It is crucial to study the suppression of these fluctuations with system size. The particularly important case of non-integrable models has been addressed so far only by numerics and conjectures based on analytical bounds. In this work, we are able to derive analytical predictions for the temporal fluctuations in a

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... able model (the transverse Ising chain with extra terms). Our results are based on identifying a dynamical regime of 'many-particle dephasing', where quasiparticles do not yet relax but fluctuations are nonetheless suppressed exponentially by weak integrability breaking.