We present detailed chemical abundance measurements for 45 globular clusters (GCs) associated with galaxies in (and, in one case, beyond) the Local Group. The measurements are based on new high-resolution integrated-light spectra of GCs in the galaxies NGC 185, NGC 205, M 31, M 33, and NGC 2403, combined with reanalysis of previously published observations of GCs in the Fornax dSph, WLM, NGC 147, NGC 6822, and the Milky Way. The GCs cover the range −2.8 < [Fe/H] < −0.1 and we determined abundances for Fe, Na, Mg, Si, Ca, Sc, Ti, Cr, Mn, Ni, Cu, Zn, Zr, Ba, and Eu. Corrections for non local thermodynamic equilibrium effects are included for Na, Mg, Ca, Ti, Mn, Fe, Ni, and Ba, building on a recently developed procedure. For several of the galaxies, our measurements provide the first quantitative constraints on the detailed composition of their metal-poor stellar populations. Overall, the GCs in different galaxies exhibit remarkably uniform abundance patterns of the α, iron-peak, and neutron-capture elements, with a dispersion of less than 0.1 dex in [α/Fe] for the full sample. There is a hint that GCs in dwarf galaxies are slightly less α-enhanced (by ∼0.04 dex on average) than those in larger galaxies. One GC in M 33 (HM33-B) resembles the most metal-rich GCs in the Fornax dSph (Fornax 4) and NGC 6822 (SC7) by having α-element abundances closer to scaled-solar values, possibly hinting at an accretion origin. A principal components analysis shows that the α-element abundances strongly correlate with those of Na, Sc, Ni, and Zn. Several GCs with [Fe/H] < −1.5 are deficient in Mg compared to other α-elements. We find no GCs with strongly enhanced r-process abundances as reported for metal-poor stars in some ultra-faint dwarfs and the Magellanic Clouds. The similarity of the abundance patterns for metal-poor GCs in different environments points to similar early enrichment histories and only allow for minor variations in the initial mass function.