Searches for the lepton number violating $K^{+}\rightarrow\pi^{-}\mu^{+}e^{+}$ decay and lepton flavour violating decays $K^{+}\rightarrow\pi^{+}\mu^{-}e^{+}$ and $\pi^{0}\rightarrow\mu^{-}e^{+}$ are reported, using NA62 data collected in 2017 and 2018. In the absence of a significant excess above the background in each channel, upper limits on the branching ratios are established at $90\%$ confidence level: $\mathcal{B}(K^{+}\rightarrow\pi^{-}\mu^{+}e^{+})<4.2\times10^{-11}$, $\mathcal{B}(K^{+}\rightarrow\pi^{+}\mu^{-}e^{+})<6.6\times10^{-11}$, and $\mathcal{B}(\pi^{0}\rightarrow\mu^{-}e^{+})<3.2\times10^{-10}$.