  The goal of this work is to find the simplest UV completion of Accidental
Composite Dark Matter Models (ACDM) that can dynamically generate an asymmetry
for the DM candidate, the lightest \textit{dark baryon} (DCb), and
simultaneously annihilate the symmetric component. In this framework the DCb is
a bound state of a confining $\text{SU}(N)_{\text{DC}}$ gauge group, and can
interact weakly with the visible sector. The constituents of the DCb can
possess non-trivial charges under the Standard Model gauge group. The
generation of asymmetry for such candidate is a two-flavor variation of the
\emph{out-of-equilibrium} decay of a heavy scalar, with mass $M_\phi\gtrsim
10^{10}$ GeV. Below the scale of the scalars, the models recover accidental
stability, or long-livedness, of the DM candidate. The symmetric component is
annihilated by residual confined interactions provided that the mass of the DCb
$m_{\text{DCb}} \lesssim 75$ TeV. We implement the mechanism of asymmetry
generation, or a variation of it, in all the original ACDM models, managing to
generate the correct asymmetry for DCb of masses in this range. For some of the
models found, the stability of the DM candidate is not spoiled even considering
generic GUT completions or asymmetry generation mechanisms in the visible
sector.
