We present the results of medium‐resolution spectroscopy of 28 globular clusters (GCs) in six nearby galaxies of different luminosities and morphological types, situated in M33 (15 objects), M31 (three), IC10 (four), UGCA86 (four), Holmberg IX (one) and DDO71 (one) obtained at the Special Astrophysical Observatory 6‐m telescope. Measurements of Lick absorption line indices and comparison with Simple Stellar Population models enabled us to obtain their spectroscopic ages, metallicities and α‐element to Fe abundance ratios. We found that all old‐ and intermediate‐age GCs in our sample have low metallicities [Z/H]≲ ‐0. 8 dex. Metal‐rich clusters are young and are preferentially found in galaxies more massive than ∼ 109 M⊙. The least massive dwarfs of our sample, DDO71 and Holmberg IX, host one massive intermediate‐age and one massive young metal‐poor GC, respectively. [α/Fe] abundance ratios tend to be enhanced but closer to solar values for dwarf galaxies compared to GCs in more massive galaxies. We analyse the age–metallicity relation for GCs in our sample and others from the literature, and find that (1) there is a general trend for GCs in low surface brightness dwarf galaxies to be more metal‐poor at a given age than GCs in more massive galaxies; (2) the GC metallicity spread is wider for more massive galaxies and (3) intermediate‐age GCs in early‐type dwarf galaxies are more metal‐rich at any given age than those in irregular galaxies of similar luminosity.