  The hierarchical mergers that form the haloes of dark matter surrounding
galaxies, groups and clusters are not entirely efficient, leaving substantial
amounts of dense substructure, in the form of stripped halo cores or
`subhaloes', orbiting within these systems. Using a semi-analytic model of
satellite dynamics, we study the evolution of haloes as they merge
hierarchically, to determine how much substructure survives merging and how the
properties of individual subhaloes change over time. We find that subhaloes
evolve, due to mass loss, orbital decay, and tidal disruption, on a
characteristic time-scale equal to the period of radial oscillations at the
virial radius of the system. Subhaloes with realistic densities and density
profiles lose 25-45 per cent of their mass per pericentric passage, depending
on their concentration and on the circularity of their orbit. As the halo
grows, the subhalo orbits also grow in size and become less bound. Based on
these general patterns, we suggest a method for including realistic amounts of
substructure in semi-analytic models based on merger trees. We show that the
parameters in the resulting model can be fixed by requiring self-consistency
between different levels of the merger hierarchy. In a companion paper, we will
compare the results of our model with numerical simulations of halo formation.
