Two approaches are presented for extending the FETI-DP domain-decomposition-based iterative method to the solution of problems with Linear Multipoint Constraints (LMPCs). Both cases of arbitrary LMPCs and LMPCs generated by the mortar method for enforcing a weak continuity of the solution across non-matching finite element interfaces are considered. In the first approach, the LMPCs are addressed during the preconditioning step but enforced only at convergence. In the second approach, the LMPCs are enforced at each iteration through the solution of an auxiliary coarse problem. Both extended FETI-DP solvers are benchmarked on a parallel processor for a wide range of constrained structural mechanics problems. The obtained results reveal that both solvers are numerically scalable and complement each other from the memory and CPU performance viewpoints.