Suppose that $M$ is compact $N$ is simply connected and has finitely generated homology, and the codimension $n-m$ is at least $3$. Then the space $Emb(M,N)$ is such that

(1) for every basepoint $\pi_1$ is solvable and $\pi_k$ is finitely generated for all $k\ge 1$. 

This can be proved using the Weiss tower and the "analyticity" or "multiple disjunction" result of John Klein and myself. Or it is possible to give an argument that does not use the tower. Either way, you repeatedly use the fact that when (1) holds for the base of a fibration and for every fiber then it also holds for the total space, and the fact that if (1) holds for the base and the total space of a fibration then it holds for every fiber.