Let $\Theta$ be an $\mathbb R^d$-valued Gaussian-mixture random variable with $k$ Gaussian components $\Phi_1,\Phi_2,\ldots,\Phi_k$, where $\Phi_i$ has mean vector $\mu_i$ and covariance matrix $\Sigma_i$, and mixing probabilities $p_1,p_2,\ldots,p_k$. Let $C$ be the conditioning categorical random variable. Then, $Y_x=\Theta^Tx$ is a random process indexed by $x\in\mathbb R^d$. For any $x\in\mathbb R^d$, the moment-generating function of $\Theta^Tx$ is
$$ \begin{aligned} M_{\Theta^Tx}(t)=\mathbf E[e^{t\Theta^Tx}]=\mathbf E[\mathbf E[e^{t\Theta^Tx}|C]]=\sum_{i=1}^kp_iM_{\Phi_i^Tx}(t) \end{aligned} $$
Since linear transformations of Gaussian random variables are Gaussian, $\Phi_i^Tx$ has mean $\mu_i^Tx$ and variance $x^T\Sigma_ix$. Therefore, $\Theta^Tx$ is Gaussian-mixture with $k$ components, the $i$-th component having mean $\mu_i^Tx$ and variance $x^T\Sigma_ix$, and the same mixing probabilities $p_1,p_2,\ldots,p_k$.
Can $Y_x$ be called a Gaussian-mixture process or a mixture-Gaussian process or a mixture of Gaussian processes? Has this kind of random process been studied?