The Equation of State (EoS) of asymmetric nuclear matter has been explored through the study of mid-rapidity fragment dynamics from the 35 MeV/u $^{70}$Zn $^{70}$Zn, $^{64}$Zn $^{64}$Zn, and $^{64}$Ni $^{64}$Ni systems. The experimental data was collected at the Texas A and M Cyclotron Institute using the 4 NIMROD-ISiS array, which provided both event characterization and excellent isotopic resolution of charged particles. The transverse collective flow was extracted for proton, deuteron, triton, 3He, alpha, and 6He particles. Isotopic and isobaric effects were observed in the transverse flow of the fragments. In both cases, the transverse flow was shown to decrease with an increasing neutron content in the fragments. The (N/Z)sys dependence of the transverse flow and the difference betwen the triton and 3He flow were shown to be sensitive to the density dependence of the symmetry energy using the stochastic mean-field model. A stiff parameterization of Esym(p) was found to provide better agreement with the experimental data. The transverse flow for intermediate mass fragments (IMFs) was investigated, providing a new probe to study the nuclear EoS. A transition from the IMF flow strongly depending on the mass of the system, in the most violent collisions, to a dependence on the charge of the system, for the peripheral reactions, was observed. Theoretical simulations were used to show that the relative differences in the IMF flow are sensitive to the density dependence of the symmetry energy. The best agreement between the experiment and theory was achieved with a stiff Esym(p). A new method was developed in which correlations between the projectile-like and mid-rapidity fragments were examined using a scaled flow. Theoretical simulations were used to show that the scaled flow of the particles was connected to their average order of emission. The experimental results suggest that the mid-rapidity region is preferentially populated with neutron-rich light charged particles and the Z=3-4 IMFs at a relatively early stage in the collision. This work presents additional constraints on the nuclear EoS and insight into the mid-rapidity dynamics observed in Fermi energy heavy-ion collisions.