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The transition form factors for doubly heavy baryons into a spin-$1/2$ or spin-$3/2$ ground-state baryon induced by both the charged current and the flavor changing neutral current are systematically studied within the light-front quark model. In the transition the two spectator quarks have two spin configurations and both are considered in this calculation. We use an updated vertex functions, and inspired by the flavor SU(3) symmetry, we also provide a new approach to derive the flavor-spin factors. With the obtained transition form factors, we perform a phenomenological study of the corresponding semi-leptonic decays of doubly heavy baryons induced by the $c\to d/s \ell^+ν$, $b\to c/u\ell^-\bar ν$ and $b\to d/s\ell^+\ell^-$. Results for partial decay widths, branching ratios and the polarization ratios $Γ_{L}/Γ_{T}$s are given. We find that most branching ratios for the semi-leptonic decays induced by the $c\to d,s$ transitions are at the order of $10^{-3}\sim10^{-2}$, which might be useful for the search of other doubly-heavy baryons. Uncertainties in form factors, the flavor SU(3) symmetry and sources of symmetry breaking effects are discussed. We find that the SU(3) symmetry breaking effects could be sizable in charmed baryon decays while in the bottomed case, the SU(3) symmetry breaking effects are less significant. Our results can be examined at the experimental facilities in the future.