  We define canonical refinements of Harder-Narasimhan filtrations and
stratifications in moduli theory, generalising and relating work of
Haiden-Katzarkov-Kontsevich-Pandit and Kirwan. More precisely, we define a
canonical stratification on any noetherian algebraic stack $\mathcal X$ with
affine diagonal that admits a good moduli space and is endowed with a norm on
graded points. The strata live in a newly defined stack of sequential
filtrations of $\mathcal X$. Therefore the stratification gives a canonical
sequential filtration, the iterated balanced filtration, for each point of
$\mathcal X$. Examples of applicability include moduli of principal bundles on
a curve, moduli of objects at the heart of a Bridgeland stability condition and
moduli of K-semistable Fano varieties. We conjecture that the iterated balanced
filtration describes the asymptotics of the Kempf-Ness flow in Geometric
Invariant Theory, as part of a larger project aiming to describe the
asymptotics of natural flows in moduli theory. In the case of quotient stacks
by diagonalisable algebraic groups, we give an explicit description of the
iterated balanced filtration in terms of convex geometry.
