  We consider the Brownian SYK, i.e. a system of $N$ Majorana (Dirac) fermions
with a white-noise $q$-body interaction term. We focus on the dynamics of the
Frame potentials, a measure of the scrambling and chaos, given by the moments
of the overlap between two independent realisations of the model. By means of a
Keldysh path-integral formalism, we compute its early and late-time value. We
show that, for $q>2$, the late time path integral saddle point correctly
reproduces the saturation to the value of the Haar frame potential. On the
contrary, for $q=2$, the model is quadratic and consistently we observe
saturation to the Haar value in the restricted space of Gaussian states
(Gaussian Haar). The latter is characterised by larger system size corrections
that we correctly capture by counting the Goldstone modes of the Keldysh saddle
point. Finally, in the case of Dirac fermions, we highlight and resolve the
role of the global $U(1)$ symmetry.
