Magnetic chirality is an important knob in spintronics and can be engineered through structural symmetry breaking of magnetic thin film multilayers. The dynamics of chiral domain walls is determined by the cooperation of chiral contributions in the magnetic energy functional as well as in the dissipation tensor which need to be better controlled for the sake of the device applications. In this work, we performed a systematic study of magnetic field-induced magnetic bubble expansion in structural inversion asymmetric multilayers with different Pt thicknesses using polar magneto-optical Kerr microscopy. Asymmetric expansion of magnetic bubble is investigated in the creep regime as a function of in-plane and out-of-plane magnetic fields. The results reveal the competition between two key mechanisms governing the asymmetry in the field-driven domain wall expansion, namely the Dzyaloshinskii-Moriya interaction and the chiral magnetic damping. The interplay between these two effects leads to the seemingly counterintuitive experimental signature, depending on the strength of the external magnetic field. The effective control on the bubble asymmetry expansion can be of great importance for future memory and multiplexer-based applications.