We report magnetic susceptibility, high-field magnetization, muon spin relaxation, and electron spin resonance measurements of the breathing pyrochlore antiferromagnets LiGa1-xInxCr4O8 (x=0.2, 0.5). Unlike the previously proposed spin-glass-like phase for 0.1<x<0.75, we find no signature for spin glassiness and phase segregation in both the x=0.2 and 0.5 compounds. Instead, we identify a two-step magnetic transition with a partial spin freezing at T∗=12K (x=0.2) and 9 K (x=0.5) followed by quasistatic order at Tm=6K (x=0.2) and 3 K for (x=0.5). In addition, for Tm<T<T∗, we observe evidence of a competition between fast and slow spin dynamics, suggesting a thermal and temporal distribution of spin correlations. Our findings underscore the possibility of realizing novel magnetic phases by tuning bond alternation and introducing bond disorder through chemical substitution.