We have studied the onset process of dynamic transformation of the internal structure of a moving domain wall (DW) in soft magnetic thin film nanostrips, driven by applied magnetic fields larger than the Walker field strength, Hw. It was found that one of the edge-soliton cores of a transverse wall (TW)-type DW should reach a critical nucleation size of the vortex core by moving inward beyond a critical deviation, Δ ycri, in the transverse (width) direction, in order for the transformation from a TW to a vortex wall (VW) (or antivortex wall, AVW) to occur. The value of Δ ycri is estimated to be close to the full width at half maximum of the out-of-plane magnetizations of a stabilized vortex core (or antivortex core). Upon completion of the nucleation of the vortex (antivortex) core, the VW (AVW) is stabilized, accompanying its characteristic gyrotropic motion in a potential well (hill) of a given nanostrip. Field strengths exceeding the Hw, the onset field of DW velocity breakdown, are not sufficient but necessary conditions for the dynamic DW transformations.