We investigate the evolutionary connection between local IR-bright galaxies ($\log L_{\rm IR}\ge 11.4\,L_\odot$) and quasars. We use high angular resolution ($\sim$ 0.3-0.4 arcsec $\sim$ few hundred parsecs) $8-13\,\mu$m ground-based spectroscopy to disentangle the AGN mid-IR properties from those of star formation. The comparison between the nuclear $11.3\,\mu$m PAH feature emission and that measured with Spitzer/IRS indicates that the star formation is extended over a few kpc in the IR-bright galaxies. The AGN contribution to the total IR luminosity of IR-bright galaxies is lower than in quasars. Although the dust distribution is predicted to change as IR-bright galaxies evolve to IR-bright quasars and then to optical quasars, we show that the AGN mid-IR emission of all the quasars in our sample is not significantly different. In contrast, the nuclear emission of IR-bright galaxies with low AGN contributions appears more heavily embedded in dust although there is no clear trend with the interaction stage or projected nuclear separation. This suggests that the changes in the distribution of the nuclear obscuring material may be taking place rapidly and at different interaction stages washing out the evidence of an evolutionary path. When compared to normal AGN, the nuclear star formation activity of quasars appears to be dimming whereas it is enhanced in some IR-bright nuclei, suggesting that the latter are in an earlier star-formation dominated phase.