We present X-ray spectral variability of 24 local active galactic nuclei (AGN) from the Palomar sample of nearby galaxies, as observed mainly by Swift. From hardness ratio measurements, we find that 18 AGN with low accretion rates show hardening with increasing count rate, converse to the softer-when-brighter behaviour normally observed in AGN with higher accretion rates. Two AGN show softening with increasing count rate, two show more complex behaviour, and two do not show any simple relationship.Sufficient data were available for the spectra of 13 AGN to be summed in fluxbins. In 9 of these sources, correlated luminosity-dependent changes in the photon index (Γ) of a power-law component are found to be the main cause of hardness variability. For 6 objects, with a low accretion rate as a fraction of the Eddington rate (ṁ Edd ), Γ is anticorrelated withṁ Edd , i.e. 'harder-when-brighter' behaviour is observed. The 3 higher-ṁ Edd -rate objects show a positive correlation between Γ anḋ m Edd . This transition from harder-when-brighter at lowṁ Edd to softer-when-brighter at highṁ Edd can be explained by a change in the dominant source of seed-photons for X-ray emission from cyclo-synchrotron emission from the Comptonising corona itself to thermal seed-photons from the accretion disc. This transition is also seen in the 'hard state' of black hole X-ray binaries (BHXRBs). The results support the idea that LINERs are analogues of BHXRBs in the hard state and that Seyferts are analogues of BHXRBs in either the high-accretion-rate end of the hard state or in the hard-intermediate state.