The early X-ray afterglow of gamma-ray bursts revealed by Swift carried many surprises. We focus in this paper on the plateau phase whose origin remains highly debated. We confront several newly discovered correlations between prompt and afterglow quantities (isotropic emitted energy in gamma-rays, luminosity and duration of the plateau) to several models proposed for the origin of plateaus in order to check if they can account for these observed correlations. We first show that the scenario of plateau formation by energy injection into the forward shock leads to an efficiency crisis for the prompt phase and therefore study two possible alternatives: the first one still takes place within the framework of the standard forward shock model but allows for a variation of the microphysics parameters to reduce the radiative efficiency at early times; in the second scenario the early afterglow results from a long-lived reverse shock. Its shape then depends on the distribution of energy as a function of Lorentz factor in the ejecta. In both cases, we first present simple analytical estimates of the plateau luminosity and duration and then compute detailed light curves. In the two considered scenarios we find that plateaus following the observed correlations can be obtained under the condition that specific additional ingredients are included. In the forward shock scenario, the preferred model supposes a wind external medium and a microphysics parameter epsilon_e that first varies as n^{- u} (n being the external density), with u~1 to get a flat plateau, before staying constant below a critical density n_0. To produce a plateau in the reverse shock scenario the ejecta must contain a tail of low Lorentz factor with a peak of energy deposition at Gamma >~ 10.
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