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The realization and detection of two-leg ladders with the interplay of the artificial gauge field provide an ideal platform for investigations of the quantum phase transition and quantum dynamics. Here, we realize the spatio-temporal evolution dynamics of two-leg ladder systems with magnetic flux analytically and numerically. The quench dynamics induced by giving sudden quenches of initial density distributions of left and right leg components, rung-to-leg coupling ratios and magnetic flux from the ground state are discussed in detail. Affected by the chirality of ladder systems together with the tunneling between the two legs, rich and interesting quench dynamics observed, which including the Josephson Oscillations (JOs), Chiral Density Wave (CDW), Scissors Mode (SM), Periodic Torsional Vortex (PTV) and Periodic Torsional-Scissors Vortex (PTSV). Remarkably, we find that the oscillation frequency of the quench dynamics associated with coupling ratios and magnetic flux can be used to identify the phase transition between the Meissner phase and Vortex phase. Therefore, our work may provide an effective way to distinguish phase transitions by using quench dynamics. [ABSTRACT FROM AUTHOR]