Ergodicity on the Profinite Integers versus Ergodicity on the $p$-adic integers

Question

Let $\mathbb{P}$ denote the set of prime numbers, let $\mathbb{Z}_{p}$ denote the ring of $p$-adic integers, and let: $$\widetilde{\mathbb{Z}}\overset{\textrm{def}}{=}\prod_{p\in\mathbb{P}}\mathbb{Z}_{p}$$denote the ring of profinite integers. Using the haar probability measures on the $\mathbb{Z}_{p}$s, one can define a haar probability measure on $\widetilde{\mathbb{Z}}$ via the product measure construction. Now, let $q$ be an integer $\geq2$, and define:$$\widetilde{\mathbb{Z}}_{\textrm{off }q}\overset{\textrm{def}}{=}\prod_{\begin{array}{c} p\in\mathbb{P}\\ p\nmid q \end{array}}\mathbb{Z}_{p}$$ Next, letting $b\in\mathbb{Z}$, consider the formula $f\left(x\right)=qx+b$. I can show that $f$ then defines an ergodic map on $\mathbb{Z}_{p}$ for all $p\nmid q$. My intuition tells me that this should then force $f$ to be an ergodic map on $\widetilde{\mathbb{Z}}_{\textrm{off }q}$, but I don't know how to prove it.