I have a question about pushforward maps for the motivic cohomology groups $H^p(X, \mathbf{Q}(q))$, for $X$ a smooth variety over a characteristic 0 field.

According to Mazza--Voevodsky--Weibel "Lectures on motivic cohomology", these groups have the following *covariant* functoriality properties (in addition to their usual contravariant functoriality)

if $f: X \to Y$ is a proper flat morphism, with fibres of dimension $d$, there is a pushforward map $f_*: H^{p + 2d}(X, \mathbf{Q}(q + d)) \to H^{p}(Y, \mathbf{Q}(q))$;

if $f: X \to Y$ is the inclusion of a closed subvariety of codimension $c$, then there is a pushforward map $f_*: H^{p}(X, \mathbf{Q}(q)) \to H^{p + 2c}(Y, \mathbf{Q}(q + c))$.

(Note that the first map lowers the cohomological degree, while the second map raises it.)

There are also corresponding maps in $\ell$-adic étale cohomology (for any prime $\ell$), or in Beilinson's absolute Hodge cohomology when the base field is $\mathbf{R}$ or $\mathbf{C}$; and there are "realisation" maps from motivic cohomology to these other two cohomology theories.

Are the pushforward maps (1) and (2) known to be compatible with their étale and Hodge analogues under the realisation maps?

I guess this must be known, but I've been unable to find a reference which states this directly. For étale cohomology, the work of Ciskinski--Deglise "Etale motives" apparently gives a realisation functor (on some category of motivic complexes) that is "compatible with Grothendieck's six operations", which sounds promising, but I'm not sufficiently expert to know if the concrete compatibility statements I want follow from this. And I've been unable to find any reference treating the case of absolute Hodge cohomology.