We use deep, spatially resolved spectroscopy from the Large Early Galaxy Astrophysics Census Survey to study radial variations in the stellar population of 17 spectroscopically selected post-starburst (PSB) galaxies. We use spectral fitting to measure two Lick indices, H δ_A and Fe 4383 , and find that, on average, PSB galaxies have radially decreasing H δ_A and increasing Fe 4383  profiles. In contrast, a control sample of quiescent, non-PSB galaxies in the same mass range shows outwardly increasing H δ_A and decreasing Fe 4383 . The observed gradients are weak (≈−0.2 Å/R_e), mainly due to seeing convolution. A two-SSP (simple stellar population) model suggests that intrinsic gradients are as strong as observed in local PSB galaxies (≈−0.8 Å/R_e). We interpret these results in terms of inside-out growth (for the bulk of the quiescent population) versus star formation occurring last in the centre (for PSB galaxies). At z ≈ 0.8, central starbursts are often the result of gas-rich mergers, as evidenced by the high fraction of PSB galaxies with disturbed morphologies and tidal features (40 per cent). Our results provide additional evidence for multiple paths to quiescence: a standard path, associated with inside-out disc formation and with gradually decreasing star formation activity, without fundamental structural transformation, and a fast path, associated with centrally concentrated starbursts, leaving an inverse age gradient and smaller half-light radius.