Context. Globular clusters (GCs) are emerging as powerful tracers of the chemical composition of extragalactic stellar populations. Aims. We present new abundance measurements for eleven GCs in the Local Group galaxies NGC 147, NGC 6822, and Messier 33. These are combined with previously published observations of four GCs in the Fornax and WLM galaxies. Methods. The abundances were determined from analysis of integrated-light spectra, obtained with the HIRES spectrograph on the Keck I telescope and with UVES on the VLT. We used our analysis technique that has been developed for this purpose and tested on Milky Way GCs. Results. We find that the clusters with [Fe/H] < −1.5 are all α-enhanced at about the same level as Milky Way GCs. Their Na abundances are also generally enhanced relative to Milky Way halo stars, suggesting that these extragalactic GCs resemble their Milky Way counterparts in containing significant fractions of Na-rich stars. For [Fe/H] > −1.5, the GCs in M33 are also α-enhanced, while the GCs that belong to dwarfs (NGC 6822 SC7 and Fornax 4) have closer to Solar-scaled α-element abundances. The abundance patterns in SC7 are remarkably similar to those in the Galactic GC Ruprecht 106, including significantly sub-solar [Na/Fe] and [Ni/Fe] ratios. In NGC 147, the GCs with [Fe/H] < −2.0 account for about 6% of the total luminosity of stars in the same metallicity range, a lower fraction than those previously found in the Fornax and WLM galaxies, but substantially higher than in the Milky Way halo. Conclusions. At low metallicities, the abundance patterns suggest that GCs in the Milky Way, dwarf galaxies, and M33 experienced similar enrichment histories and/or processes. At higher metallicities, the lower levels of α-enhancement in the GCs found in dwarf galaxies resemble the abundance patterns observed in field stars in nearby dwarfs. Constraining the presence of multiple populations in the GCs is complicated by the lack of information about detailed abundances in field stars of the corresponding metallicities. We suggest that correlations such as [Na/Fe] vs. [Ni/Fe] may prove useful for this purpose if an accuracy of ∼ 0.1 dex or better can be reached for the integrated-light measurements.