On doubling metric measure spaces endowed with a Dirichlet form and satisfying the Davies-Gaffney estimates, we show some characterisations of pointwise upper bounds of the heat kernel in terms of one-parameter weighted inequalities which correspond respectively to the Nash inequality and to a Gagliardo-Nirenberg type nequality when the volume growth is polynomial. This yields a new and simpler proof of the well-known equivalence between classical heat kernel upper bounds and the relative Faber-Krahn inequalities. We are also able to treat more general pointwise estimates where the heat kernel rate of decay is not necessarily governed by the volume growth.