Correlations between the radio continuum, infrared, and CO emissions are known to exist for several types of galaxies and across several orders of magnitude. However, the low-mass, low-luminosity, and low-metallicity regime of these correlations is not well known. A sample of metal-rich and metal-poor dwarf galaxies from the literature has been assembled to explore this extreme regime. The results demonstrate that the properties of dwarf galaxies are not simple extensions of those of more massive galaxies; the different correlations reflect different star-forming conditions and different coupling between the star formation and the various quantities. It is found that dwarfs show increasingly weaker CO and infrared emissions for their luminosity, as expected for galaxies with a low dust content, slower reaction rates, and a hard ionizing radiation field. In the higher-luminosity dwarf regime [L1.4 GHz ≳ 1027 W, where L1.4 GHz ≃ 1029 W for a Milky Way star formation rate (SFR) of ≃1 M⊙ yr-1], the total and non-thermal radio continuum emissions appear to adequately trace the SFR. A breakdown of the dependence of the (H α-based) thermal, non-thermal, and, hence, total radio continuum emission on SFR occurs below L1.4 GHz ≃ 1027 W, resulting in a steepening or downturn of the relations at extreme low luminosity. Below LFIR ≃ 1036 W ≃ 3 × 109 L⊙, the infrared emission ceases to adequately trace the SFR. A lack of a correlation between the magnetic field strength and the SFR in low SFR dwarfs suggests a breakdown of the equipartition assumption. As extremely metal-poor dwarfs mostly populate the low SFR and low-luminosity regime, they stand out in their infrared, radio continuum, and CO properties.