$\newcommand{\Diff}{\mathrm{Diff}} \newcommand{\Emb}{\mathrm{Emb}} \newcommand{\Homeo}{\mathrm{Homeo}} \newcommand{\HomEq}{\mathrm{HomEq}}$This is an ancient question, but I suppose it has many answers. 

Yes. 

There's quite a few tools. 

For example, consider the group of diffeomorphisms of the $n$-disc that restrict to the identity on the boundary.  Call this $\Diff(D^n)$.   There is a 'scanning map' 

$$\Diff(D^n) \to \Omega^i \Emb(D^{n-i}, D^n)$$

defined (heuristically) by considering $D^n$ to be a product $D^n = D^i \times D^j$ with $i + j = n$.  i.e. we are considering the disc to be fibered by parallel $j$-dimensional discs, in an $i$-dimensional family. 

So if you take the induced map on homotopy groups you get a map

$$\pi_k \Diff(D^n) \to \pi_{i+k} \Emb(D^j, D^n).$$

The above map in the $i=1$ case is (what I call) Cerf's *scanning* homotopy-equivalence.

$$\Diff(D^n) \simeq \Omega \Emb(D^{n-1}, D^n).$$

In general not much is known about these maps when $i>1$.  That said, these maps are currently something that are being actively studied.

On the other hand, if you change the manifold a little bit and look at $\Diff(S^1 \times D^n)$ then there are scanning maps of the form

$$\Diff(S^1 \times D^n) \to \Omega^{n-1} \Emb(I, S^1 \times D^n)$$

and these maps can be shown to be non-trivial.  That's maybe my current favorite approach, but there are certainly many other maps.  

Significantly more elementary, there are the forgetful maps $\Diff(M) \to \Homeo(M) \to \HomEq(M)$, i.e. diffeomorphisms are homeomorphisms are homotopy self-equivalences.  It can often be difficult to say things about $\Homeo(M)$, but the space of self-homotopy equivalences of a space can be studied using fairly classical tools. 

That particular map studied by Burghelea, Antonelli and Kahn, that looks like it must be an iterated composite of the connecting map in the pseudo-isotopy fibration sequence for $\Diff(D^n)$.   That's a fairly specialized tool.  That said, the diffeomorphism group of an arbitrary manifold fits into a pseudo-isotopy fiber sequence, and so the connecting map there is perhaps the most directly analogous to your $L$.  Specifically, it is the tautological map $\Omega \Diff(M) \to \Diff(I \times M)$, i.e. one is identifying $\Omega \Diff(M)$ with the subgroup of diffeomorphisms of $I \times M$ that preserve the height in the $I$ parameter.