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

We give a new proof of the $\widehat{U}$-theorem of Bérczi, 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 $\mathrm{\Theta }$-strata.