The far-field analytical expressions for the electromagnetic fields of amplitude of vector-vortex beams having a Bessel–Gauss (BG) distribution propagating in free space are obtained based on the vector angular spectrum and the method of stationary phase. The far-field energy flux distributions and the beam quality by the power in the bucket (PIB) in the paraxial and nonparaxial regimes are investigated. The PIB of the vector-vortex BG beams depend on the ratio of the waist width to wavelength and the polarization order. The analyses show that vector-vortex BG beams with low polarization order have better energy focusability in the farfield.