In recent years, the exact/inexact steepest descent algorithm for multiobjective optimization problem has been extended to Riemannian manifolds [see G. de~Carvalho~Bento, O.~P. Ferreira and P.~R. Oliveira, J. Optim. Theory Appl. {\bf 154} (2012), no.~1, 88--107; MR2931368; G. de~Carvalho~Bento, J. da~Cruz~Neto and P.~S.~M. Santos, J. Optim. Theory Appl. {\bf 159} (2013), no.~1, 108--124; MR3103289; O.~P. Ferreira, M.~S. Louzeiro and L.~F. Prudente, J. Optim. Theory Appl. {\bf 184} (2020), no.~2, 507--533; MR4054291]. \par In this paper, the authors continue to study the convergence properties of the inexact steepest descent algorithm for multiobjective optimizations on general Riemannian manifolds (without curvature constraints). The study focuses on two issues of the convergence properties: one is the full convergence, the other is the convergence rate estimate. Under the assumption of convexity (quasi-convexity), local (global) convergence results are established. Moreover, without the assumption of the local convexity (quasi-convexity), but under an error bound-like condition, local (global) convergence results and convergence rate estimates are presented, which are new even in the linear space setting. The results improve or extend the corresponding ones in [J.~H. Wang et al., SIAM J. Optim. {\bf 31} (2021), no.~1, 172--199; MR4198581] for scalar optimization problems on Riemannian manifolds to multiobjective optimization problems on Riemannian manifolds. For the special case when the inexact steepest descent algorithm employs Armijo's rule, the results improve the corresponding ones in [O.~P. Ferreira, M.~S. Louzeiro and L.~F. Prudente, op. cit.] by relaxing the curvature constraints. The approaches used in this paper are new and different from those in [G. de~Carvalho~Bento, O.~P. Ferreira and P.~R. Oliveira, op. cit.; G. de~Carvalho~Bento, J. da~Cruz~Neto and P.~S.~M. Santos, op. cit.; O.~P. Ferreira, M.~S. Louzeiro and L.~F. Prudente, op. cit.]. \par The extension of classical optimization theory and algorithms to multiobjective optimization on manifolds is an interesting topic that deserves further study.