  We use pro cdh-descent of $K$-theory to study the relationship between the
zero cycles on a singular variety $X$ and those on its desingularisation $X'$.
We prove many cases of a conjecture of S. Bloch and V. Srinivas, and relate the
Chow groups of $X$ to the Kerz--Saito Chow group with modulus of $X'$ relative
to its exceptional fibre.
