The standard classical question concerns multiplicities of the irreducible representations of $L$ (or its derived group) in the restriction: these are given by *branching rules*.   This is complicated to work out in detail but is treated in many textbooks and other sources.   I'm not sure exactly how much information you are asking for.   The original $W$-invariant set of weights relative to $T \subset L$ is unchanged, but the original representation decomposes into a direct sum of irreducibles for $L$, with the subgroup $W_L \subset W$ acting on the separate weight diagrams within the convex region you describe.   (It's easy to picture this in restricting from rank 2 to rank 1, for example.)  

By the way, your $G$-dominant weight means a weight of $T$ which is dominant relative to a fixed Borel subgroup $B$ for which $T \subset B \subset P$.