We use pro cdh-descent of -theory [the author, J. Inst. Math. Jussieu 15, No. 3, 539--567 (2016; Zbl 1366.19005)] to study the relationship between the zero cycles on a singular variety and those on its desingularisation . We prove many cases of a conjecture of S. Bloch and V. Srinivas, and relate the Chow groups of to the Kerz-Saito Chow group with modulus of relative to its exceptional fibre.