Single crystals of Yb14-xRExMnSb11 (0 < x < 0.6, RE = Pr, Nd, Sm, and Gd) were synthesized by Sn flux. The compounds are iso-structural with Ca14AlSb11 (I4(1)/acd), and their compositions were determined by wavelength-dispersive spectroscopy. Yb14MnSb11 is described as a partially screened d-metal Kondo system with the isolated [MnSb4](9-) tetrahedral cluster having a d(5) + hole configuration that results in four unpaired electrons measured in the ferromagnetically ordered phase. All of the Yb atoms in Yb14MnSb11 are present as Yb2+, and the additional RE in Yb14-xRExMnSb11 is trivalent, contributing one additional electron to the structure and altering the magnetic properties. All compounds show ferromagnetic ordering in the range of 39-52 K attributed to the [MnSb4](9-) magnetic moment. Temperature-dependent DC magnetization measurements of Yb14-xPrxMnSb11 (0.44 <= x <= 0.56) show a sharp downturn right below the ferromagnetic transition temperature. Single-crystal neutron diffraction shows that this downturn is caused by a spin reorientation of the [MnSb4](9-) magnetic moments from the ab-plane to c-axis. The spin reorientation behavior, caused by large anisotropy, is also observed for similar x values of RE = Nd but not for RE = Sm or Gd at any value of x. In Pr-, Nd-, and Sm-substituted crystals, the saturation moments are consistent with similar to 4 unpaired electrons attributed to [MnSb4](9-), indicating that local moments of Pr, Nd, and Sm do not contribute to the ferromagnetic order. In the case of RE = Pr, this is confirmed by neutron diffraction. In contrast, the magnetic measurements of RE = Gd show that the moments of Gd ferromagnetically order with the moments of [MnSb4](9-), and reduced screening of moments on Mn2+ is evident. The sensitive variation of magnetic behavior is attributed to the various RE substitutions resulting in different interactions of the 4f-orbitals with the 3d-orbitals of Mn in the [MnSb4](9-) cluster conducted through 5p-orbitals of Sb.