For $w$ a permutation, the associated **(ordinary) Schubert polynomial** $\mathfrak{S}_w(\textbf{x})$ is a multivariate polynomial that represents for the cohomology class of the Schubert variety $X_w$ in the flag variety.
There are many references for this fact. See, e.g., this text by Manivel.
One key feature of Schubert polynomials is that they satisfy a recurrence often referred to as the **Lascoux-Schutzenberger transition equation**.
Let $r$ be the largest value such that $w(r)>w(r+1)$ and $s$ be the maximum value so that $w(s) < w(r)$.
We define
$$
I(w,r) = \{i < r: w \lessdot w t_{ir}\},
$$
where $t_{rs}$ is the transposition $(r,s)$ and $\lessdot$ indicates a cover relation in Bruhat order.
Then
$$
x_r \mathfrak{S}_w(\textbf{x}) = \mathfrak{S}_{w t_{rs}}(\textbf{x}) - \sum_{i \in I(w,r)} \mathfrak{S}_{w t_{ir}}(\textbf{x}).
$$
This result dates back to work of Lascoux and Schutzenberger in the 80's and can be derived from Monk's formula.
There is a simple extension allowing one to work with arbitrary descents in $w$.

The **double Schubert polynomial** $\mathfrak{S}(\textbf{x},\textbf{y})$ is a polynomial in two alphabets of variables that represents the equivariant cohomology class of the Schubert variety $X_w$ in the flag variety.
These polynomials satisfy an analogous recurrence
$$
(x_r - y_s) \mathfrak{S}_w(\textbf{x},\textbf{y}) = \mathfrak{S}_{w t_{rs}}(\textbf{x},\textbf{y}) - \sum_{i \in I(w,r)} \mathfrak{S}_{w t_{ir}}(\textbf{x},\textbf{y}).
$$
I've seen this identity proved in a class and can rederive it myself (as well as the more general transition equations), but can't seem to find an explicit reference in the literature.
I've asked around a bit, but haven't turned anything up.
Can anyone point me to a concrete reference of the transition equations for double Schubert polynomials (or something equivalent) in the literature?

P.S. There is an unpublished preprint by Lascoux called "Chern and Yang Through Ice" that describes transition for double Grothendieck polynomials (a stronger result), but I'm wary of giving it as a reference for two reasons. First, it claims a stronger result that requires a bit of work to translate to my setting. Second, I find the whole document a great challenge to parse, and am skeptical that a complete proof is presented.