We study quench dynamics of bosonic fractional quantum Hall systems in small lattices with cylindrical boundary conditions and low particle density. The states studied have quasiholes or quasiparticles relative to the bosonic Laughlin state at half-filling. Pinning potentials are placed at edge sites (or sites close to the edges) to trap quasiholes and qausiparticles. The potentials are then turned off, and because the edges of fractional quantum Hall systems host chiral edge modes, we expect chiral dynamics of the quasiholes and quasiparticles. We numerically show that chiral motion of the density distribution is observed and robust for the case with positive potentials (quasiholes), but that there is no noticeable chiral motion for negative potentials (quasiparticles). The comparison of the numerical ground states with model lattice Laughlin wave functions suggests that both positive and negative potentials do create and pin anyons that are not necessarily well separated on small lattices. Initializing the dynamics with the model state also shows the lack of chiral dynamics of quasiparticles. Our results suggest that, in small lattices with low particle density, quasiparticles are strongly adversely affected in dynamical processes, whereas quasiholes are dynamically robust.