  The phenomenon of $B^0_q-\bar B^0_q$ mixing ($q=d,s$) provides a sensitive
probe for physics beyond the Standard Model. We explore the corresponding space
for New Physics left through the current data, having a careful look at
analyses of the Unitarity Triangle that are needed for the Standard Model
predictions of the $B_q$ mixing parameters. In particular, we explore the
impact of tensions between inclusive and exclusive determinations of the CKM
matrix elements $|V_{ub}|$ and $|V_{cb}|$. Moreover, we focus on the angle
$\gamma$ of the Unitarity Triangle, comparing measurements from $B\to DK$ and
$B\to\pi\pi$, $\rho\pi$, $\rho\rho$ decays, where the latter are typically
interpreted in terms of the angle $\alpha$. We discuss various scenarios and
present the corresponding state-of-the-art constraints on the New Physics
parameters of $B_q^0-\bar B_q^0$ mixing. We point out that these results have
an interesting application in the analysis of rare $B_q^0\to \mu^+\mu^-$
decays, allowing us to minimise the impact of CKM parameters in the search for
New Physics. In view of the high-precision era, we make future projections.
Interestingly, we find that for the extraction of the New Physics parameters in
the $B_d$ system the determination of the apex of the Unitarity Triangle
results in a key limiting factor. By contrast, the corresponding impact is
negligible for the $B_s$ system, making it a promising candidate to reveal
sources of New Physics.
