Murai observed an interesting stability pattern in the resolutions of symmetric monomial ideals. I will explain my preliminary attempt at providing a structural explanation to Murai's results by proving finiteness theorems for their local cohomology modules. Along the way, I will also prove the Le--Nagel--Nguyen--Römer conjectures for sequences of GL_n-equivariant k[x_1, x_2, ..., x_n]-modules when k is an infinite field of characteristic p > 0.