We present a derivation of the linear heating Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation for a constant nucleation rate and diffusion-controlled growth, in the hard impingement approximation. The result is compared with the linear heating JMAK equation for interface-controlled growth, and with the isothermal JMAK equation. We show that all approximations made in deriving the JMAK equations (i.e. including previous work) hold when the activation energies involved are large compared to the thermal energy, which turns out to be virtually always the case. Finally, we demonstrate in a simple way that within the JMAK framework, peak shift methods such as Kissinger analysis, Marseglia and Ozawa plots are formally equivalent and may all be used to analyse experimental data.