This thesis presents the results of a time-dependent analysis of $B^0\to D^{\mp}\pi^{\pm}$ decays using $3~\rm fb^{-1}$ of proton-proton collision data collected with the LHCb detector at CERN's Large Hadron Collider during Run 1 with a centre-of-mass energy of $7$ (2011) and $8$ (2012) TeV. The LHCb experiment is dedicated to the study of the properties of $b$-flavoured hadrons, in particular $CP$ violation in the $B$ meson system. The Standard Model of Particle Physics describes very precisely the mechanism and the amount of $CP$ violation expected in the Universe. However, the observed matter-antimatter asymmetry is larger by several order of magnitude compared to the predictions. This could be explained by the existence of a new source of $CP$ violation, originating in New Physics beyond the Standard Model. The time-dependent analysis of $B^0\to D^{\mp}\pi^{\pm}$ decays provides constraints on the angle $\gamma$ of the Unitarity Triangle, one of the fundamental parameters of the Standard Model related to $CP$ violation. Since no sizeable high-order Standard Model processes are expected to contribute, any deviation from the predictions would be an unambiguous signature of New Physics. The current experimental precision on $\gamma$ is significantly lower than that of theoretical predictions. This motivates the effort for new experimental determinations of $\gamma$ in order to reduce its uncertainty. The analysis of $\Bz\to\Dmp\pipm$ decays, although not as sensitive as that obtained from decays of charged $B$ mesons into $D^{(*)0}K^{(*)+}$ final states, represents an independent and uncorrelated estimation of $\gamma$ that contributes to the global combination of all $\gamma$ measurements. The result obtained in this thesis is more precise than previous determinations from other experiments (BaBar, Belle) using $B^0\to D^{\mp}\pi^{\pm}$ decays. Although based on a very large sample of about half a million signal events, it is still dominated by statistical uncertainties, indicating good prospects for future improvements in precision with more data from Run 2 and beyond. In addition to the $B^0\to D^{\mp}\pi^{\pm}$ analysis, this thesis also summarizes the studies to improve the performances of the flavour tagging algorithms used by the LHCb collaboration to infer the flavour of neutral $B$ mesons in time-dependent analyses. The performance of these algorithms, being correlated with the kinematics of the reconstructed particles as well as the complexity of the event (number of tracks and primary vertices), showed a significant decrease on Run 2 data (2015--2018), which were collected at a centre-of-mass energy of $13~\rm TeV$. Thanks to new implementations, these algorithms now have a performance similar to that obtained with Run 1 data.