In this Letter, we investigate the binding mechanism and motion dynamics of the bound state consisting of two pure-quartic solitons (PQSs) with unequal intensities and find that their movement occurs as an entity under the Raman self-frequency shift. By calculating the forces that induce the relative motion between the unequal PQSs, we derive the balanced conditions for maintaining a near-constant separation and the constant phase profile between them. The predictions are validated by the numerical simulations. Our work provides insights into the intrinsic features of symmetry-broken localized structures in nonlinear-dispersive systems, stimulating interest in asymmetric multi-soliton states with intriguing dynamics.