  Gravitationally-bound clusters that survive gas removal represent an unusual
mode of star formation in the Milky Way and similar spiral galaxies. While
forming, they can be distinguished observationally from unbound star formation
by their high densities, virialised velocity structures, and star formation
histories that accelerate toward the present, but extend multiple free-fall
times into the past. In this paper we examine several proposed scenarios for
how such structures might form and evolve, and carry out a Bayesian analysis to
test these models against observed distributions of protostellar age, counts of
young stellar objects relative to gas, and the overall star formation rate of
the Milky Way. We show that models in which the acceleration of star formation
is due either to a large-scale collapse or a time-dependent increase in star
formation efficiency are unable to satisfy the combined set of observational
constraints. In contrast, models in which clusters form in a "conveyor belt"
mode where gas accretion and star formation occur simultaneously, but the star
formation rate per free-fall time is low, can match the observations.
