The Fourier transform of a compactly supported smooth function on Lie groups over $\mathbb{Q}_S$, where $S$ contains finitely many primes and $\infty$

Question

Let $G$ be a semisimple Lie group defined over global Field $\mathbb{Q}$. Let $S$ be a set of finitely many non-Archimedean places including Archimedean places. Let $P_{0}=M_{0}A_{0}N_{0}$ be the minimal parabolic subgroup with Langlands decompostion defined over $\mathbb{Q}$. We consider the $\mathbb{Q}_{S}$ points of $P_{0}$ and name those subgroups as $M_{0,S}$, $A_{0,S}$, and $N_{0,S}$. Suppose $K_{S}=K_{\infty}\prod\limits_{v<\infty} K_{v} $ be the maximal compact subgroup. Let $C_{c}^{\infty}(G(\mathbb{Q}_{S}),K_{\infty})$ be space of compactly supported smooth functions, which are bi-$K_{\infty}$-finite at Archimedean place. Let $\mathfrak{a}_{0,S}=\text{Lie}(A_{0,S}/A_{0,S}\cap K_{S})$ and $\nu \in \mathfrak{a}_{0,S ,\mathbb{C}}^*$, the complexified dual of $\mathfrak{a}_{0,S}$.
Spherical Case: When we consider the smooth compactly supported bi-$K_{S}$ -invariant functions f, we can define the Fourier transform as composition of Satake isomorphism and Fourier transform on Abelian group:
$$\hat{f}(\nu)=\int\limits_{A_{0,S}}\int\limits_{N_{0,S}}f(an)e^{(-\nu+\rho)(\ln(a))}dnda.$$
Bi-$K_{\infty}$-case: Let us only consider function on $G(\mathbb{R})$. Let $(\tau,V_{\tau})$ be an irreducible representation of $K_{\infty}$. We consider the functions f which are bi-$K_{\infty}$-finite with respect to $\tau$. Let us denote this space of functions as $C_{c}^{\infty}(G(\mathbb{R}),K_{\infty})$. Let $\sigma_{\infty}$ be an irreducible admissible representation of $M_{0,\infty}$, such that $\sigma_{\infty}\subset \tau|_{M_{0,\infty}}$. Let $\lambda_{\infty} \in \mathfrak{a}_{0,\infty,\mathbb{C}}^*$. Let $\mathcal{P}(M_{0,\infty})$ be the set of parabolic subgroups whose Levi part is $M_{0,\infty}A_{0,\infty}$. Let us denote the normalized principle series representation for $B \in \mathcal{P}(M_{0,\infty})$ as
$$\text{Ind}_{B}^{G(\mathbb{R})}(\sigma_{\infty}\otimes e^{-\lambda_{\infty}+\rho})=\pi_{B}(\sigma_{\infty},\lambda_{\infty}).$$ Then by Arthur's Paley-Wiener theorem at Archimedean place we can define the operator valued Fourier Transform as the following family:
$$\{\pi_{B}(\sigma_{\infty},\lambda_{\infty})(f): B \in \mathcal{P}(M_{0,\infty}),\sigma_{\infty},\lambda_{\infty}\}$$ Non-Archimedean case: Let $f$ be a locally constant compactly supported function on $G(\mathbb{Q}_{v})$, for $v\ne \infty$. Let $P_{v}=M_{v}N_{v}$ be the levi decomposition of a standard parabolic containing the minimal parabolic $P_{0,v}$. Let $\sigma_{v}$ be the irreducible representation of the Levi subgroup $M_{v}$. Let $X(M_{v})$ be the group of $\mathbb{Q}_{v}$ rational characters of $M_{v}$. Let $\mathfrak{a}_{M,v,\mathbb{C}}^*=X(M_{v})\otimes \mathbb{C}$. We can define the following induced representation with parameters $\sigma_{v},\lambda_{v}$:
$$\text{Ind}_{P_{v}}^{G(\mathbb{Q}_{v})}(\sigma_{v}\otimes e^{\lambda_{v}})=\pi_{P_{v}}(\sigma_{v},\lambda_{v}).$$ Let $\mathcal{F}(M_{0,v})$ be the set of parabolic subgroups whose Levi component contains $M_{0,v}$. Hence we can define the Fourier transform of $f$ via the trace Paley-Wiener theorem by Bernstein, Deligne and Kazdhan as: $$\{\text{Tr}\pi_{P_{v}}(\sigma_{v},\lambda_{v})(f): P_{v}\in \mathcal{F}(M_{0,v}),\sigma_{v},\lambda_{v}\}$$ Question: How can I combine these two definition of Fourier transform to get the Fourier transform of $f \in C_{c}^{\infty}(G(\mathbb{Q}_{S}),K_{\infty})$? Is it going to be same as the global induced operator?
Thank you for your patience in reading the question. I apologize if this is long read in setting up the question.