The time behavior of bright spatial solitons in biased photorefractive media is investigated within the framework of a bidimensional band transport model. Biasing the photorefractive media requires an externally applied electric field or the presence of a photovoltaic effect. These two basically different phenomena are shown to be equivalent and additive. The mechanism of space-charge field buildup is analytically expressed, leading to a time-dependent wave propagation equation in generic photorefractive media. The temporal behavior of the soliton solutions to this equation is investigated. It shows the evolution of the soliton beam from the time the external electric field is applied to the final steady-state soliton. On the way, the so-called quasisteady soliton is retrieved, along with its properties. Furthermore, the photovoltaic soliton is described by the wave propagation equation: its behavior is the same as that of the steady-state soliton, the transient states included. Finally, low-power photorefractive bright spatial solitons are generated in a Bi12TiO20 crystal with a He-Ne laser and their temporal behavior is investigated, thus providing an experimental validation of our theoretical considerations.