How to use ReplaceAll iteratively over a list of lists?

I want to build a matrix J for each of the solutions of an equation.And each solution is also a set of 3 variables (x,y,z).

In the end, I want to calculate the eigenvalues of J for J applied to each of the solutionsin the solution set.

I know how to apply the matrix J for a given solution,

J/.{ x -> a, y -> b, z-> c }

if the solution was (x,y,z)=(a,b,c)

However, the solution is, for instance,

{{x -> a, y -> b, z -> c}, {x -> d, y -> e, z -> f}}

So I want to loop over the solution list and apply J for each specific solution.How can I do that in an automated way?The simpler, the better... I don't to have to really "code" in Mathematica.

Here is my real problem:

I start from the function

F[x_, y_, z_, a_, b_, c_, d_] := (1 - a + b - (b/3) (c x + d (y + z))) x

which generates this set of fixed points (each element of the list FP is a solution)

FP = Solve[{x == F[x, y, z, a, b, c, d], y == F[y, x, z, a, b, c, d], z == F[z, x, y, a, b, c, d]}, {x, y, z}]

Then I build the Jacobian matrix:

J = FullSimplify[  {{D[F[x, y, z, a, b, c, d], x], D[F[x, y, z, a, b, c, d], y], D[F[x, y, z, a, b, c, d], z]},   {D[F[y, x, z, a, b, c, d], x], D[F[y, x, z, a, b, c, d], y], D[F[y, x, z, a, b, c, d], z]},   {D[F[z, x, y, a, b, c, d], x], D[F[z, x, y, a, b, c, d], y], D[F[z, x, y, a, b, c, d], z]}}  ]

I can calculate the eigenvalues of J applied to the first solution like this:

Eigenvalues[J/.FP[[1,All]]]

But how do I do that iteratively, generating another list?

Thanks