Consider a sequence of conditional pdf's $p_n(y | x)$ on a Polish space $X \times Y$, endowed with its Borel sigma algebra. Suppose, as $n\rightarrow \infty$, in $\mathbb{L}_1$ ($N$ denotes the normal distribution):

$$ \sum_k \omega_k (X) N(\beta_{0,k} + \beta_{1,k} X, \sigma^2_k) = p_n(y | X) \stackrel{\mathbb{L}_1}{\rightarrow} p_0(y | X) = \sum_k \omega^*_k (X) N(\beta^*_{0,k} + \beta^*_{1,k} X, \sigma^2_{k,*}) $$

is then true that for a sequence of measurable function $f_n \stackrel{\mathbb{L}_1}{\rightarrow} f_0$ also the "perturbed" conditional expectations converge as follows:

$$ \sum_k \omega_k (X) (\beta_{0,k} + \beta_{1,k} (X+f_n(X))) \stackrel{\mathbb{L}_1}{\rightarrow} \sum_k \omega^*_k (X) (\beta^*_{0,k} + \beta^*_{1,k} (X+f_0(X))) $$

PS: this question is related to this, but it is not the same, as I believe it is harder.