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We investigate the quench dynamics of a two-component Bose mixture and study the onset of modulational instability, which leads the system far from equilibrium. Analogous to the single-component counterpart, this phenomenon results in the creation of trains of bright solitons. We provide an analytical estimate of the number of solitons at long times after the quench for each of the two components based on the most unstable mode of the Bogoliubov spectrum, which agrees well with our simulations for quenches to the weak attractive regime when the two components possess equal intraspecies interactions and loss rates. We also explain the significantly different soliton dynamics in a realistic experimental homonuclear potassium mixture in terms of different intraspecies interaction and loss rates. We investigate the quench dynamics of the particle number of each component estimating the characteristic time for the appearance of modulational instability for a variety of interaction strengths and loss rates. Finally, we evaluate the influence of the beyond-mean-field contribution, which is crucial for the ground-state properties of the mixture, in the quench dynamics for both the evolution of the particle number and the radial width of the mixture. In particular, even for quenches to strongly attractive effective interactions, we do not observe the dynamical formation of solitonic droplets.