  Hot and warm Jupiters (HJs and WJs correspondingly) are gas-giants orbiting
their host stars at very short orbital periods ($P_{HJ}<10$ days;
$10<P_{WJ}<200$ days). HJs and a significant fraction of WJs are thought to
have migrated from an initially farther-out birth locations. While such
migration processes have been extensively studied, the thermal evolution of
gas-giants and its coupling to the the migration processes are usually
overlooked. In particular, gas-giants end their core-accretion phase with large
radii and then contract slowly to their final radii. Moreover, intensive
heating can slow the contraction at various evolutionary stages. The initial
inflated large radii lead to faster tidal migration due to the strong
dependence of tides on the radius. Here we explore this accelerated migration
channel, which we term inflated eccentric migration, using a semi-analytical
self-consistent modeling of the thermal-dynamical evolution of the migrating
gas-giants, later validated by our numerical model (see a companion paper,
paper II). We demonstrate our model for specific examples and carry a
population synthesis study. Our results provide a general picture of the
properties of the formed HJs\&WJs via inflated migration, and the dependence on
the initial parameters/distributions. We show that tidal migration of
gas-giants could occur far more rapidly then previously thought and lead to
accelerated destruction and formation of HJs and enhanced formation rate of
WJs. Accounting for the coupled thermal-dynamical evolution is therefore
critical to the understanding of HJs/WJs formation, evolution and final
properties of the population and play a key role in their migration process.
