If \begin{equation} \langle f \rangle = \int\int f(x_{A}, x_{B})\text{d}P_{A}(x_{A})\,\text{d}P_{B}(x_{B}|x_{A}) \end{equation} then, assuming Fubini's theorem holds (for instance assuming $f(\cdot,\cdot)\ge 0$) \begin{align} \langle f \rangle &= \int\left\{\int f(x_{A}, x_{B})\text{d}P_{B}(x_{B}|x_{A})\right\}\text{d}P_{A}(x_{A})\\ &=\int\left\{\int f(x_{A}, x_{B})\frac{\text{d}P_B(x_B|x_A)}{\text{d}P_B^{\rm eff}(x_B)}\text{d}P_{A}(x_{A})\right\}\text{d}P_{B}(x_{B})\\ &=\int\underbrace{\left\{\int f(x_{A}, x_{B})\text{d}P_{A}(x_{A}|x_{B})\right\}}_{\mathbb E[f(X_A,x_B)|x_B]}\text{d}P_{B}^{\rm eff}(x_{B})\\ \end{align} which clearly differs from $$\int\left\{\int f(x_{A}, x_{B}) \text{d}P_{A}(x_{A})\right\}\text{d}P_{B}^{\rm eff}(x_{B})$$

If \begin{equation} \langle f \rangle = \int\int f(x_{A}, x_{B})\text{d}P_{A}(x_{A})\,\text{d}P_{B}(x_{B}|x_{A}) \end{equation} then, assuming Fubini's theorem holds (for instance assuming $f(\cdot,\cdot)\ge 0$) \begin{align} \langle f \rangle &= \int\left\{\int f(x_{A}, x_{B})\text{d}P_{B}(x_{B}|x_{A})\right\}\text{d}P_{A}(x_{A})\\ &=\int\left\{\int f(x_{A}, x_{B})\frac{\text{d}P_B(x_B|x_A)}{\text{d}P_B^{\rm eff}(x_B)}\text{d}P_{A}(x_{A})\right\}\text{d}P_{B}(x_{B})\\ &=\int\underbrace{\left\{\int f(x_{A}, x_{B})\text{d}P_{A}(x_{A}|x_{B})\right\}}_{\mathbb E[f(X_A,x_B)|x_B]}\text{d}P_{B}^{\rm eff}(x_{B})\\ \end{align} which clearly differs from $$\int\left\{\int f(x_{A}, x_{B}) \text{d}P_{A}(x_{A})\right\}\text{d}P_{B}^{\rm eff}(x_{B})$$ How much they differ will depend on the choice of the function $f$ and the dependence between $X_A$ and $X_B$. Considering $$\sup_{f;\ \langle f\rangle=1}\left| \int\int f(x_{A}, x_{B}) \text{d}P_{A}(x_{A})\text{d}P_{B}^{\rm eff}(x_{B}) - 1\right|$$ gives a measure of dependence between $X_A$ and $X_B$ connected with notions like $\alpha$-mixing, $\beta$-mixing and other measures of dependence.

If \begin{equation} \langle f \rangle = \int\int f(x_{A}, x_{B})\text{d}P_{A}(x_{A})\,\text{d}P_{B}(x_{B}|x_{A}) \end{equation} then, assuming Fubini's theorem holds (for instance assuming $f(\cdot,\cdot)\ge 0$) \begin{align} \langle f \rangle &= \int\left\{\int f(x_{A}, x_{B})\text{d}P_{B}(x_{B}|x_{A})\right\}\text{d}P_{A}(x_{A})\\ &=\int\left\{\int f(x_{A}, x_{B})\frac{P_B(x_B|x_A)}{P_B^{\rm eff}(x_B)}\text{d}P_{A}(x_{A})\right\}\text{d}P_{B}(x_{B})\\ &=\int\underbrace{\left\{\int f(x_{A}, x_{B})\text{d}P_{A}(x_{A}|x_{B})\right\}}_{\mathbb E[f(X_A,x_B)|x_B]}\text{d}P_{B}^{\rm eff}(x_{B})\\ \end{align} which clearly differs from $$\int\left\{\int f(x_{A}, x_{B}) \text{d}P_{A}(x_{A})\right\}dP_{B}^{\rm eff}(x_{B})$$

If \begin{equation} \langle f \rangle = \int\int f(x_{A}, x_{B})\text{d}P_{A}(x_{A})\,\text{d}P_{B}(x_{B}|x_{A}) \end{equation} then, assuming Fubini's theorem holds (for instance assuming $f(\cdot,\cdot)\ge 0$) \begin{align} \langle f \rangle &= \int\left\{\int f(x_{A}, x_{B})\text{d}P_{B}(x_{B}|x_{A})\right\}\text{d}P_{A}(x_{A})\\ &=\int\left\{\int f(x_{A}, x_{B})\frac{\text{d}P_B(x_B|x_A)}{\text{d}P_B^{\rm eff}(x_B)}\text{d}P_{A}(x_{A})\right\}\text{d}P_{B}(x_{B})\\ &=\int\underbrace{\left\{\int f(x_{A}, x_{B})\text{d}P_{A}(x_{A}|x_{B})\right\}}_{\mathbb E[f(X_A,x_B)|x_B]}\text{d}P_{B}^{\rm eff}(x_{B})\\ \end{align} which clearly differs from $$\int\left\{\int f(x_{A}, x_{B}) \text{d}P_{A}(x_{A})\right\}\text{d}P_{B}^{\rm eff}(x_{B})$$