  We investigate the high-$p_T$ tails of the $pp\to \ell \nu$ and $pp \to \ell
\ell$ Drell-Yan processes as probes of New Physics in semileptonic interactions
with an arbitrary flavor structure. For this purpose, we provide a general
decomposition of the $2\to2$ scattering amplitudes in terms of form-factors
that we match to specific scenarios, such as the Standard Model Effective Field
Theory (SMEFT), including all relevant operators up to dimension-$8$, as well
as ultraviolet scenarios giving rise to tree-level exchange of new bosonic
mediators with masses at the TeV scale. By using the latest LHC run-II data in
the monolepton ($e\nu$, $\mu\nu$, $\tau\nu$) and dilepton ($ee$, $\mu\mu$,
$\tau\tau$, $e\mu$, $e\tau$, $\mu\tau$) production channels, we derive
constraints on the SMEFT Wilson coefficients for semileptonic four-fermion and
dipole operators with the most general flavor structure, as well as on all
possible leptoquark models. For the SMEFT, we discuss the range of validity of
the EFT description, the relevance of $\mathcal{O}(1/\Lambda^2)$ and
$\mathcal{O}(1/\Lambda^4)$ truncations, the impact of $d=8$ operators and the
effects of different quark-flavor alignments. Finally, as a highlight, we
extract for several New Physics scenarios the combined limits from high-$p_T$
processes, electroweak pole measurements and low-energy flavor data for the
$b\to c\tau\nu$ transition, showing the complementarity between these different
observables. Our results are compiled in {\tt HighPT}, a package in {\tt
Mathematica} which provides a simple way for users to extract the Drell-Yan
tails likelihoods for semileptonic effective operators and for leptoquark
models.
