Moduli spaces for $\Theta$-strata and non-reductive quotients

We give a new proof of the $\hat{U}$-theorem of B\'erczi, Doran, Hawes and Kirwan on the existence of geometric quotients for actions of graded unipotent groups in terms of stacks of filtrations and gradings introduced by Halpern-Leistner. Our proof works over any affine Noetherian base, in particular it simultaneously generalizes the previous results to arbitrary characteristic, actions in families and to general $\Theta$-strata.