In their 1983 paper "Remarks on correspondences and algebraic cycles" Bloch and Srinivas proved several interesting properties of smooth proper varieties (over universal domains) whose Chow groups of zero cycles are supported in positive codimension. Now there exist a lot of papers citing this one; so I would like to ask the following questions.
Which texts describe 'the biggest' collections of varieties satisfying this support condition (or possibly some alternative one)?
Did anybody study similar support conditions for (co)homology theories distinct from $Chow_0$ (computed over a universal domain)? This could include: $Chow_0$ (with integral coefficients) over fields that are not algebraically closed fields, higher Chow groups, certain Witt and Chow-Witt groups, K-theory.
What interesting consequences are known to follow from support conditions of this type?