A number of Ln(2)O(2)MSe(2) (Ln = La and Ce; M = Fe, Zn, Mn, and Cd) compounds, built from alternating layers of fluorite-like [Ln(2)O(2)](2+) sheets and antifluorite-like [MSe2](2-) sheets, have recently been reported in the literatures. The available MSe4/2 tetrahedral sites are half-occupied, and different compositions display different ordering patterns: [MSe2](2-) layers contain MSe4/2 tetrahedra that are exclusively edge-sharing (stripe-like), exclusively corner-sharing (checkerboard-like), or mixtures of both. This paper reports 60 new compositions in this family. We reveal that the transition-metal arrangement can be systematically controlled by either Ln or M doping, leading to an "infinitely adaptive" structural family. We show how this is achieved in La2O2Fe1-xZnxSe2, La2O2Zn1-xMnxSe2., La2O2Mn1-xCdxSe2, Ce2O2Fe1-xZnxSe2, Ce2O2Zn1-xMnxSe2, Ce2O2Mn1-xCdxSe2, La2-yCeyO2FeSe2, La2-yCeyO2ZnSe2, La2-yCeyO2MnSe2, and La2-yCeyO2CdSe2 solid solutions.