We determine the rigid-body motions of two solid and conducting particles ${\cal P}_1$ and ${\cal P}_2$ freely suspended in a liquid metal of uniform viscosity $\mu$ and conductivity $\sigma_l>0$ when subject to uniform ambient electric and magnetic fields ${\bf E}$ and ${\bf B}.$ The translational and angular velocities ${\bf U}^{(n)}$ and \mbox{\boldmath $\Omega$}$^{(n)}$ of the particle ${\cal P}_n$ with uniform conductivity $\bsigma_n\geq 0$ are obtained without calculating the disturbed electric field and the liquid metal flow in the unbounded fluid domain. The advocated approach solely resorts to a few boundary-integral equations on the entire surface of the cluster. The work will successively establish the relevant boundary-integral equations and both propose and implement a suitable numerical strategy. Numerical results will be presented and discussed for a few two-sphere clusters of equal or inequal spheres for several settings $({\bf E},{\bf B})$ and $(\sigma_1/\sigma_l,\sigma_2/\sigma_l).$