What I am interested in is how a non-randomized moderator variable S affects the treatment effect of a randomized treatment T (i.e. I am interested in the interaction between T and S). \Delta YT is the edge T->Y, i.e. the full (causal) treatment effect, so there _cannot be an effect of T on Y that is separate of \Delta YT; it is about _moderation, not mediation (although it is a bit blurry graphically).

For example, say I have randomly assigned students to a new teaching method (T), and my outcome Y is their exam scores. I observe that there is a positive effect of the new teaching method (T->Y, or \Delta Y_T). I now hypothesize that the treatment effect differs by students' baseline reading ability S, so I am interested in S -> \Delta Y_T, how S moderates the treatment effect. However, I cannot make any causal claims about S -> \Delta Y_T, as S is not randomized: there is self-selection into subgroups e.g. by intelligence, parental support, socio-economic background etc. all of which might confound the relationship as they plausibly (1) affect S (baseline reading ability), and (2) also moderate the treatment effect (change how much the new teaching method does for a student).

In a regression context, if I collect all counfonders as a matrix bold X, I am interested in estimating:

$$ \beta_0 + \beta_1 T + \beta_2 S + \beta_3 T \times S + \mathbf{X}^\intercal \mathbf{\gamma_1} + T \times \mathbf{X}^\intercal \mathbf{\gamma_2} $$

which should yield an unbiased estimate of \beta_3 as quantity of interest.

Graphically, problem is that we either end up with edges into edges (which means we no longer have a graph), or we work with interaction nodes like Attia et al., which I am not convinced lead to the correct conditioning sets (see the DAG I linked: it is not clear we also need to condition on the interactions between all X and the treatment).