ABSTRACT We study the internal radial gradients of the stellar populations in a sample comprising 522 early-type galaxies (ETGs) from the SAMI (Sydney-AAO Multi-object Integral field spectrograph) Galaxy Survey. We stack the spectra of individual spaxels in radial bins, and derive basic stellar population properties: total metallicity ([Z/H]), [Mg/Fe], [C/Fe] and age. The radial gradient (∇) and central value of the fits (evaluated at Re/4) are compared against a set of six observables that may act as drivers of the trends. We find that velocity dispersion (σ) – or, equivalently gravitational potential – is the dominant driver of the chemical composition gradients. Surface mass density is also correlated with the trends, especially with stellar age. The decrease of ∇[Mg/Fe] with increasing σ is contrasted by a rather shallow dependence of ∇[Z/H] with σ (although this radial gradient is overall rather steep). This result, along with a shallow age slope at the massive end, imposes a substantial constraint on the progenitors of the populations that contribute to the formation of the outer envelopes of ETGs. The SAMI sample is split, by design, between ‘field’ and cluster galaxies. Only weak environment-related differences are found, most notably a stronger dependence of central total metallicity ([Z/H]e4) with σ, along with a marginal trend of ∇[Z/H] to steepen in cluster galaxies, a result that is not followed by [Mg/Fe]. The results presented here serve as stringent constraints on numerical models of the formation and evolution of ETGs.