I don't know too much about this, but I will make a couple points.
Let us assume first that $K$ is a maximal subfield of $D$. Then $D$ will be of dimension $n^2$ over $L$ and $[K:L]=n$. So viewing $D$ as a vector space over $K$ acting from the right, the left regular representation of $D$ consists of $K$-linear mappings, and we can thus realize $D$ as a subspace of $M_n(K)$. The question then becomes, will $D$ be stable under the entrywise action of $\Gamma$. In general the answer is no, but I think that it will be stable under the action of $Gal(K/L)$.
To make this a bit concrete take a look at my answer here: An example of a division ring $D$ that is **not** isomorphic to its opposite ring There $E$ has the same role as $K$ here and $F$ has the role of $L$. The extension is cyclic (and also cubic, but in general the extension of the degree does not change my points here), and you see immediately that acting by any element of $Gal(K/L)$ maps that matrix to another one of the same form, i.e. to another element of $D$. However, if you apply an element $\tau$ of $Gal(L/F)$ to all the matrix entries, you will not get the same set, if $\tau(\gamma)\neq\gamma$, where $\gamma$ is that special non-norm element. What happens in that case is that $D$ is mapped to another division algebra $D^\tau$. In other words $\tau$ acts on the Brauer group $Br(K/L)\le Br(L)$. I'm sure this action can be described also at the level of Hasse invariants (if $K,L,F$ are number fields).
The case of cyclic division algebra is nothing special here. That element $\gamma$ there just defines a corresponding 2-cocycle of $Gal(K/L)$ acting on $L^*$. In the case of a non-cyclic extension $K/L$ the desription of a 2-cocycle is more complicated, and my copy of Jacobson's Basic Algebra II is in my office, so I will pass that. Similar phenomenons occur, though. The 2-cocycle may be stable or may not, if we let an element of $Gal(K/F)$ act on it, and again $D^\tau$ may be a different element of $Br(K/L)$.
If $K$ is a subfield of $D$ but not maximal then, IIRC we can find a maximal subfield of $D$ containing $K$ that is Galois over $L$.
Similar tricks work then, I think.
Two related results that I found are (chapter 4 of Jacobson's Basic Algebra II):
Theorem: Let $A$ be a simple subalgebra of a finite dimensional central simple algebra $B$. Then any homomorphism of $A$ into $B$ can be extended to an inner automorphism of $B$.
Theorem: (Skolem - Noether) Any automorphism of a finite dimensional central simple algebra is inner.
I think that both these theorems assume the automorphism to be linear over the center, i.e. homomorphisms of $L$-algebras. Therefore these results do not say anything about mappings extending $Gal(L/F)$.[/edit]
On the other hand, if $K$ is very large, then I don't think we have much hope of defining an action of $Gal(K/L)$ on $D$.
I wish I could have made my answer independent of that particular representation of $D$. Hopefully some more knowledgable poster can add to this, and correct eventual errors.