  We consider families of dynamics that can be described in terms of
Perron-Frobenius operators with exponential mixing properties. For piecewise
C^2 expanding interval maps we rigorously prove continuity properties of the
drift J(l) and of the diffusion coefficient D(l) under parameter variation. Our
main result is that D(l) has a modulus of continuity of order
O(|dl||log|dl|)^2), i.e. D(l) is Lipschitz continuous up to quadratic
logarithmic corrections. For a special class of piecewise linear maps we
provide more precise estimates at specific parameter values. Our analytical
findings are verified numerically for the latter class of maps by using exact
formulas for the transport coefficients. We numerically observe strong local
variations of all continuity properties.
