Leptoquarks (LQ) with couplings to the third generation of Standard Model (SM) quarks have been proposed as possible explanations of the flavour anomalies indicating the violation of lepton flavour universality (LFV). The CMS collaboration has initiated an extensive search programme for these new states in the LHC run-2 data recorded at $\sqrt{s}=13$ TeV. In this article, the CMS search results in the LQ pair-production final states $\tau b\,\tau b$, $\nu t\,\nu t$, $\nu b\,\nu b$, $\tau t\,\tau t$, and $\mu t\,\mu t$, as well as in the LQ single-production final state $\tau\,\tau b$ are discussed. No significant deviation from the SM is observed in any of these channels. For a broad range of LQ decay modes, exclusion limits on the LQ masses are determined at 95% confidence level reaching from 1.0 TeV to 1.8 TeV.