Dendrite growth is a micro-scale phenomenon ubiquitously observed in both natural and industrial settings, in which the evolution of microstructure is influenced significantly by non-equilibrium physical phenomena, such as melt convection, heat and mass transfer, and solid-phase movement and decisively impact and determine the macroscopic mechanical properties of the resultant structures. This study introduces a coupled lattice Boltzmann-phase field (LB-PF) model to simulate dendritic growth and motion during non-equilibrium solidification involving solute diffusion and melt flow. In the present model, the anisotropic lattice Boltzmann scheme is employed to describe the growth of dendrite, while the two-relaxation-time (TRT) lattice Boltzmann scheme is utilized to compute melt flow. The MLS (Mei-Luo-Shyy) method manages the fluid-structure interaction between the dendrite and the melt, and the Galilean invariant momentum exchange method is used to calculate the hydrodynamic force and moment acting on dendrite.The model is quantitatively validated by the simulation of the continuous growth and the drafting-kissing-tumbling (DKT) phenomenon. Subsequently, it is applied to investigate the impact of dendrite movement and interfacial non-equilibrium on evolution of dendritic patterns for Si-9.0at$\%$As and the columnar to equaxied transition (CET) for Al-3.0wt$\%$Cu alloys. Investigations into the growth and remelt processes of isolated dendrites reveal that that the growth, remelting and movement of dendrite fragments greatly affect the microstructure evolution and solute segregation patterns. At low cooling rates, the solute trapping effect increases the primary dendrite arm spacing (PDAS), while the solute drag effect is opposite. At high cooling rates, the solute trapping and solute drag will make the solid-liquid interface unstable and the solidification structure develops into seaweed crystals. In addition, the results show that the timing of equiaxed dendrites nucleation and the interface non-equilibrium effects exert a significant influence on the occurrence and final morphology of the CET. This model inherits the parallelism of lattice Boltzmann method and the thermodynamic self-consistency of phase field method.