We apply a four-body version of the continuum-discretized coupled-channels method (four-body CDCC) to $^6$Li elastic scattering and investigate four-body dynamics ($n+p+\alpha+\mathrm{T}$, where $\mathrm{T}$ is a target). Four-body CDCC reproduces the measured cross sections without introducing any adjustable parameter for $^{6}$Li + $^{209}$Bi scattering at 24-50 MeV and $^{6}$Li + $^{208}$Pb scattering at 29-210 MeV. We confirm that $^6$Li breakup effects are important in a wide energy range, and then focus on the competition between four-body ($^6\mathrm{Li}+\mathrm{T}\leftrightarrow n+p+\alpha+\mathrm{T}$) and three-body ($^6\mathrm{Li}+\mathrm{T}\leftrightarrow d+\alpha+\mathrm{T}$) channel-couplings. As an interesting property, we find that $^6$Li is mainly broken up into two particles $d$ and $\alpha$,
and hardly into three particles $n$, $p$ and $\alpha$ during scattering. This property is now called "$d\alpha$-dominance". Following the $d\alpha$-dominance, we propose an effective three-body model that simulates four-body CDCC calculations reasonably well.