  We present a comprehensive study of the electroweak phase transition in
composite Higgs models, where the Higgs arises from a new, strongly-coupled
sector which confines near the TeV scale. This work extends our study in Ref.
[1]. We describe the confinement phase transition in terms of the dilaton, the
pseudo-Nambu-Goldstone boson of broken conformal invariance of the composite
Higgs sector. From the analysis of the joint Higgs-dilaton potential we
conclude that in this scenario the electroweak phase transition can naturally
be first-order, allowing for electroweak baryogenesis. We then extensively
discuss possible options to generate a sufficient amount of CP violation -
another key ingredient of baryogenesis - from quark Yukawa couplings which vary
during the phase transition. For one such an option, with a varying charm quark
Yukawa coupling, we perform a full numerical analysis of tunnelling in the
Higgs-dilaton potential and determine regions of parameter space which allow
for successful baryogenesis. This scenario singles out the light dilaton region
while satisfying all experimental bounds. We discuss future tests. Our results
bring new opportunities and strong motivations for electroweak baryogenesis.
