Proton transfer along the hydrogen bonds of DNA can lead to the creation of short-lived, but biologically relevant point mutations that can further lead to gene mutation and, potentially, cancer. In this work, the energy landscape of the canonical A-T and G-C base pairs (standard, amino-keto) to tautomeric A*-T* and G*-C* (non-standard, imino-enol) Watson-Crick DNA base pairs is modelled with density functional theory and machine-learning nudge-elastic band methods. We calculate the energy barriers and tunnelling rates of hydrogen transfer between and within each base monomer (A, T, G and C). We show that the role of tunnelling in A-T tautomerisation is statistically unlikely due to the presence of a small reverse reaction barrier. On the contrary, the thermal populations of the G*-C* point mutation could be non-trivial and propagate through the replisome. For the direct intramolecular transfer, the reaction is hindered by a substantial energy barrier. However, our calculations indicate that tautomeric bases in their monomeric form have remarkably long lifetimes.