Given a linear system $A\textbf{x}=\textbf{b}$, we can express it into the easier-to-solve right-preconditioned form:
$$ AM^{-1}\textbf{y}=\textbf{b}, \quad \textbf{y}= M^{-1}\textbf{x} $$
On the other hand, the left-preconditioned system is:
$$ M^{-1}A\textbf{x}=M^{-1}\textbf{b} $$
From the literature we know it's easy to show that the left-preconditioned linear system above is equivalent to a fixed-point linear iteration as:
$$ \textbf{x}^{(k+1)}=(I-M^{-1}A)\textbf{x}^{(k)}+M^{-1}\textbf{b} $$
Or equivalently:
$$ \textbf{x}^{(k+1)}=\textbf{x}^{(k)} + M^{-1}\textbf{r}^{(k)} $$
My question is: how can I get an update scheme like those starting from the right-preconditioning formulation?
You can just apply Richardson to the right-preconditioned system,
$\mathbf{y}_{k+1} = \mathbf{y}_k + (\mathbf{b} - AM^{-1}\mathbf{y}_k) = (I - AM^{-1})\mathbf{y}_k + \mathbf{b}$,
and on convergence you solve $\mathbf{x} = M^{-1}\mathbf{y}$.