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The Babcock-Leighton (BL) flux-transport model is a widely-accepted dynamo model of the Sun. This dynamo model has been extensively studied in a two-dimensional (2D) mean-field framework in both kinematic and non-kinematic regimes. Recent three-dimensional (3D) models have been restricted to the kinematic regime. In these models, the surface poloidal flux is produced by the emergence of bipolar magnetic regions (BMRs) that are tilted according to Joy's law. We investigate the prescription for emergence of a BMR in 3D non-kinematic simulations. We also report initial results of cyclic BL dynamo simulation. We extend a conventional 2D mean-field model of the BL flux-transport dynamo into 3D non-kinematic regime. The large-scale mean flows are driven by the parameterized $Λ$-effect in this model. For the induction equation, we use a BL source term by which the surface BMRs are produced in response to the dynamo-generated toroidal field inside the convection zone. We find that, in the 3D non-kinematic regime, the tilt angle of a newly-emerged BMR is very sensitive to the prescription for the subsurface structure of the BMR. Anti-Joy tilt angles are found unless the BMR is deeply embedded in the convection zone. We also find that the leading spot tends to become stronger than the following spot. The anti-Joy's law trend and the morphological asymmetry of the BMRs can be explained by the Coriolis force acting on the Lorentz-force-driven flows. Furthermore, we demonstrate that the solar-like magnetic cycles can be successfully obtained if the Joy's law is explicitly given in the BL $α$-effect. In these cyclic dynamo simulation, a strong Lorentz force feedback leads to cycle modulations in the differential rotation and meridional circulation. The non-axisymmetric components of the flows are found to exist as inertial modes such as the equatorial Rossby modes.