The spin-orbit (SO) interactions in low-lying electronic states of the LiM (M = Na, K, Rb, Cs) molecular series are studied through ab initio calculations of potential energy curves and SO coupling matrix elements as functions of the interatomic distance, R. Two different approaches are employed: (a) the Fock-space relativistic coupled-cluster calculations (FS-RCC) which directly yield full relativistic energies, Urel(R); the SO coupling functions, ξso(R), are extracted a posteriori through projecting scalar-relativistic wave functions onto the subspaces spanned by their full-relativistic counterparts; (b) the evaluation of the scalar-relativistic electronic energies, Usr(R), and relevant ξso(R) functions using the configuration interaction method with core-valence correlation accounted for using core polarization potentials (CI-CPP). The SO-free potentials and SO coupling functions obtained within the framework of both approaches are in good agreement with each other and their prior theoretical and empirical counterparts.