Long-time (~1 ns region) molecular dynamics (MD) simulations of lithium metasilicate (Li₂SiO₃) and a mixed-alkali silicate (LiKSiO₃) glass have been performed to confirm the mechanism of the `mixed-alkali effect'. The motion of lithium ions in lithium metasilicate (Li₂SiO₃) glass is divided into slow (A) and fast (B) categories in the glassy state. The waiting time distribution of the jump motion of each component shows a power-law behaviour with different exponents. The slow dynamics is caused by localized jump motions and by the long waiting time. On the other hand, the fast dynamics of the lithium ions in Li₂SiO₃ is characterized as Lévy flight caused by cooperative jumps. Short intervals between jump events also occur in fast dynamics, in the short-time region. The main diffusion and conduction processes of silicate glasses are not the single jumps but the cooperative jumps. A component with accelerated dynamics is almost absent in the mixed-alkali system. The contributions of the temporal and spatial aspects of the particle dynamics are separated. A large change in the spatial parameters has been observed on mixing, and interception of the jump path by other kinds of ion path suppresses the cooperative jump process. On the other hand, in longer-time regions, motion of the framework has been found to accompany the small number of events where alkali metal ions jump to unlike-ion sites. Thus both loosening of the glass structure and a reduction of ionic diffusion coefficients occur in the mixed-alkali system.