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.