With the discovery of the doubly heavy $$\Xi _{cc}$$ ${\Xi }_{\mathrm{cc}}$ baryon, comprehensive studies of the properties of the doubly heavy baryons are started. In the present work, we examine the $$\Xi _{bb} \rightarrow \Sigma _b \ell ^+ \ell ^-$$ ${\Xi }_{\mathrm{bb}}\to {\Sigma }_b{\ell }^{+}{\ell }^{-}$ and $$\Xi _{cc} \rightarrow \Sigma _c \ell ^+ \ell ^-$$ ${\Xi }_{\mathrm{cc}}\to {\Sigma }_c{\ell }^{+}{\ell }^{-}$ decays induced by flavor-changing neutral currents (FCNC) in the framework of the light-cone sum rules. After obtaining the sum rules for the form factors induced by the tensor current, the branching ratios of the relevant transitions are estimated. We found that the branching ratio for the $$c \rightarrow u$$ $c\to u$ transition is around five orders smaller than the $$b \rightarrow d$$ $b\to d$ transition. Our findings are also compared with other approaches.