Let $G$ be an affine algebraic group over an algebraically closed field $k$ of zero characteristic. The set of cosets $X_G=G(k((t))/G(k[[t]])$ is called the Affine Grassmannian of $G$ and can be given the structure of an ind-$k$-variety, so that for a closed embedding of groups $H\to G$, we get a natural morphism $X_H\to X_G$, which is a closed embedding if $G/H$ is affine. I am interested in a (as detailed as possible) reference for this, but from a specific perspective as will be explain in the following.

There are several possible (equivalent) constructions. A direct approach, in the language of ind-varieties, can be found in Kumar's "Infinite grassmannians and Moduli spaces of G-bundles" for example (for a reductive $G$, which is fine for me), which describes the ind-variety structure explicitly. Apart of using some representation theory, that I don't know well enough, it also lacks an explicit universal property which makes it difficult to operate with and in particular to construct morphsims to and from it.

A more abstract approach is to describe a functor $\operatorname{Gr}_G:kAlg\to Set$ for which $X_G$ is the set of $k$-points. Here is were the $G$-bundles (torsors) appear. There are the "global" and "local" approaches, in which roughly, $\operatorname{Gr}_G (A)$ classifies $A$-families of $G$-Bundles on a curve or a formal disc resp. together with a trivializaion away from a point. Now that one has a functor, it is possible to show that it is an ind-scheme. It is this approach that I would like to have a reference for.

I would like to mention, that this approach is outlined in Gaitsgory's seminar notes, and a more competent student would probably be able to fill in the details by herself, but unfortunately I find it difficult, so I was hopping there might be a more thorough treatment available.