Magnetic refrigeration technology is considered as a better alternative to traditional steam compression scheme, since it has many advantages such as environment friendly characteristic, more compact solid refrigerant, low cost, etc. The mechanical stability is of essential importance for serving as magnetic refrigerant materials which work under repeatedly thermal and magnetic cycles. Recent experiment reveals that the polycrystalline FeMnP1-xSix compounds are brittle, and even fracture of samples during post heat treatment is observed. Therefore, the improvement of the ductility of Fe2P-Type FeMn-based magnetocaloric materials becomes an important issue in practical application. So far, there are few studies of the mechanical properties of these compounds. Alloying is an effective method to improve the mechanical properties of single phase materials, and Ga or Ge could be a better choice to replace the Si element. In this paper, we study the mechanical properties of giant magnetocaloric FeMnP1-xTx (T=Si, Ga, Ge) compounds by the projector augmented wave method as implemented in VASP (Vienna ab initio simulation package) in the framework of density functional theory. It is found that the lattice parameter, total energy, magnetic moment, elastic constant and the electronic structure of FeMnP1-xGax compounds are similar to those of FeMnP1-xGex compounds, therefore, it is believed that the FeMnP1-xGax compounds are candidate refrigerant for room temperature magnetic refrigeration. The relatively large single crystalline elastic constants of FeMnP1-xTx (T=Si, Ga, Ge) compounds show that this family of compounds is mechanically stable. This ensures the long-term applicability of FeMnP1-xTx compounds in magnetic refrigeration facilities. For polycrystalline compounds, we calculate their shear moduli and bulk moduli by Hill averaging scheme. And according to Pugh criterion, the ductility or brittleness characteristics of FeMnP1-xTx (T=Si, Ga, Ge) compounds are discussed. All the FeMnP0.67T0.33 (T=Si, Ga, Ge) compounds are ductile, among them, FeMnP0.67Ga0.33 compound shows the best ductility, whereas the ductility of FeMnP0.67Si0.33 compound is the weakest. This result proves that substituting P with Ga could improve the ductility of this class of compound. The mechanical properties of polycrystalline FeMnP0.33T0.67 compounds are close to the ductile/brittle critical point. For FeMnP0.33T0.67 compounds, the T atoms just occupy the 2c sites of metalloid atom in Fe2P-type structure, therefore it is expected that the occupation disorders of P and T atoms at high T concentration could improve the ductility of the compounds according to the result of FeMnP0.67Ga0.33 compound. Finally, the self-consistent elastic constants of different compounds are understood from the calculated electronic density of states and force theorem.