The last solar minimum activity period, and the consequent minimum modulation conditions for cosmic rays, was unusual. The highest levels of galactic protons were recorded To simulate this effect, the rigidity dependence of the diffusion coefficients had to decrease significantly below ~3 GeV. The modulation minimum period of 2009 can thus be described as relatively more 'diffusion dominated' than previous solar minima. However, we illustrate that drifts still had played a significant role but that the observable modulation effects were not as well correlated with the waviness of the heliospheric current sheet as before. Protons still experienced global gradient and curvature drifts as the heliospheric magnetic field had decreased significantly until the end of 2009, in contrast to the moderate decreases observed during previous minimum periods. We conclude that all modulation processes contributed to the observed increases in the proton spectra for this period, exhibiting an intriguing interplay of these major mechanisms.