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The mixing of neutral mesons has been playing a key role in the formulation and testing of the Standard Model. It is also sensitive to some of the highest Beyond the Standard Model scales probed in laboratory experiments. In light of the planned LHCb Upgrades, Belle~II and its possible upgrade, and the broad interest in flavour physics in the tera-$Z$ phase of the proposed FCC-$e e$ program, I discuss constraints on New Physics contributions to $B_d$ and $B_s$ mixing which can be obtained in these benchmark scenarios. Then, I discuss the leading-order renormalization of double-insertions of dimension-6 four-fermion operators that change quark flavor by one unit (i.e., $ | \Delta F | = 1 $, $ F = $ strange-, charm-, or bottom-flavor), by dimension-8 operators relevant to meson mixing (i.e., $ | \Delta F | = 2 $) in SMEFT, setting then bounds on the Wilson coefficients of operators of dimension-6.