The influence of intense laser fields can cause asymmetries in a target system, either by breaking its symmetry or by suppressing or amplifying its intrinsic asymmetries. These asymmetries allow for coherent control of molecular processes. In this work, we present the numerical investigations that explore the effects of such asymmetries on the fragmentation dynamics of diatomic molecules (specifically, H2+ and HeH+), with a particular focus on the mid-IR pulses. By time-resolved monitoring of the dissociation processes, we observe the mechanism of directional electron localization and tunnel ionization through the asymmetry induced by the phase-controlled two-color pulse in the "symmetric" H2+ system, and the mechanism of "ladder-climbing" pathways in the HeH+ system. We also propose a visualization method for molecular orbitals through the analysis of photoelectron spectra obtained through tunnel ionization investigations. Additionally, we have studied the scarcely researched mass-asymmetry dependent isotope effects in strong fields, which show significant dependence of the dissociation probabilities on the mass-asymmetry.