In this paper, an asymmetrically curved microchannel device is designed and fabricated to quantitatively characterize the dynamic inertial focusing process of polystyrene particles and blood cells flowing along the channel. The experimental investigations are systematically carried out to probe into the regulation mechanisms of flow rate and particle size. Specifically, based on the particle fluorescent streak images and the corresponding intensity profiles at specific downstream positions, the lateral migration behaviors of particles in the mirochannel can be divided into two stages: the formation of focused streak and the shift of focusing position. It is also found that the channel structures with small radii are dominant during the whole inertial focusing process. A three-stage model is then presented to elucidate the flow-rate regulation mechanism in terms of the competition between inertial lift force and Dean drag force, according to the evolution of particle focusing dynamics with increasing flow rates. By making comparisons of focusing position and focusing ratio between two different-sized particles under various experimental conditions, we find that the larger particles have better focusing performances and stabilities, and the relative focusing position of different-sized particles can be adjusted by changing the driving flow rate. Finally, the applicability of the explored inertial focusing mechanisms for manipulating biological particles with complex features is investigated by analyzing the lateral migration behaviors of blood cells in the asymmetrically curved microchannel. The obtained conclusions are very important for understanding the particle focusing dynamics in micro-scale flows and developing the point-of-care diagnostic instruments.