  Light-cone sum rules for the $\Lambda_b \to p, \Lambda$ transition form
factors are derived from the correlation functions expanded by the twist of the
distribution amplitudes of $\Lambda_b$ baryon. In terms of the $\Lambda_b$
three-quark distribution amplitudes models constrained by the QCD theory, we
calculate the form factors at small momentum transfers and compare the results
with that estimated in the conventional light-cone sum rules (LCSR) and
perturbative QCD approaches. Our results indicate that the two different
version of sum rules can lead to the consistent numbers of form factors
responsible for $\Lambda_b \to p$ transition. The $\Lambda_b \to \Lambda$
transition form factors from
  LCSR with the asymptotic $\Lambda$ baryon distribution amplitudes are found
to be almost one order larger than that obtained in the $\Lambda_b$-baryon
LCSR, implying that the pre-asymptotic corrections to the baryonic distribution
amplitudes are of great importance. Moreover, SU(3) symmetry breaking effect
between the form factors $f_1^{\Lambda_b \to p}$ and $f_1^{\Lambda_b \to
\Lambda}$ are computed as $28^{+14}_{-8} %$ in the framework of
$\Lambda_b$-baryon LCSR.
