We denote $\|\cdot\|_F$ as the Frobenius norm of some matrix. We define $f: \mathbb{R}^{d\times r}\times\mathbb{R}^{d\times r} \rightarrow \mathbb{R}^{d\times r}$ as the following: \begin{align} f(A,B) = AR_{A,B}-B,\text{where }R_{A,B} = \arg\min_{R\in \mathbb{R}^{r\times r}, RR^\top = I}\|AR - B\|_F^2. \end{align} Now we have three matrices $X,Y$ and $Z$, where $\|X\|_F, \|Y\|_F$ and $\|Z\|_F$ can be bounded by some constant $C$. We want to find the relation between $\|f(X,Y)\|_F$ and $\|f(X,Z) - f(Y,Z)\|_F$. It is easy to see that \begin{align} \|f(X,Z) - f(Y,Z)\|_F \geq \|f(X,Y)\|_F, \end{align} because \begin{align} \|f(X,Z) - f(Y,Z)\|_F &= \|XR_{X,Z} - Z - (YR_{Y,Z} - Z)\|_F \\ & = \|XR_{X,Z}-YR_{Y,Z} \|_F \\ & = \|XR_{X,Z}R_{Y,Z}^\top - Y\|_F\\ & \geq \|XR_{X,Y} - Y\|_F\\ & = \|f(X,Y)\|_F. \end{align} The question is, is it possible to find some constant $C'$ which may related to $C$, and \begin{align} \|f(X,Z) - f(Y,Z)\|_F \leq C'\|f(X,Y)\|_F. \end{align}

PS: this problem is highly related to Orthogonal Procrustes problem. https://en.wikipedia.org/wiki/Orthogonal_Procrustes_problem