We probe the evolution of globular clusters that could form in giant molecular clouds within high-redshift galaxies. Numerical simulations demonstrate that the large and dense enough gas clouds assemble naturally in current hierarchical models of galaxy formation. These clouds are enriched with heavy elements from earlier stars and could produce star clusters in a similar way to nearby molecular clouds. The masses and sizes of the model clusters are in excellent agreement with the observations of young massive clusters. Do these model clusters evolve into globular clusters that we see in our and external galaxies? In order to study their dynamical evolution, we calculate the orbits of model clusters using the outputs of the cosmological simulation of a Milky Way-sized galaxy. We find that at present the orbits are isotropic in the inner 50 kpc of the Galaxy and preferentially radial at larger distances. All clusters located outside 10 kpc from the center formed in the now-disrupted satellite galaxies. The spatial distribution of model clusters is spheroidal, with a power-law density profile consistent with observations. The combination of two-body scattering, tidal shocks, and stellar evolution results in the evolution of the cluster mass function from an initial power law to the observed log-normal distribution.