Let $X$ be a topological space and $G_n(\mathbb{C}^m)$ be the space of vector subspaces of $\mathbb{C}^m$ of codimension $n$.
Let $G_n(\mathbb{C}^\infty):=\bigcup_{m=n}^{\infty}G_n(\mathbb{C}^m)$ using the natural inclusion $$G_n(\mathbb{C}^m)\hookrightarrow G_n(\mathbb{C}^{m+1})$$
Question: Is it true that $$ \lim_\stackrel{\longrightarrow}{m} [X\to G_n(\mathbb{C}^m)] \stackrel{(1)}{=} [X\to G_n(\mathbb{C^\mathbb{\infty}})]\stackrel{(2)}{=}[X\to G_n(\mathcal{H})]$$
Where the square brackets denote homotopy classes, $\lim_\stackrel{\longrightarrow}{m}$ denotes the "direct limit", and $\mathcal{H}$ is any infinite-dimensional separable Hilbert space? Is the first expression on the left-hand side called "stable-homotopy", which is weaker than homotopy?
Question: How to relate the (for instance, de Rham) cohomologies or the Chern classes of $G_n(\mathcal{H})$, $G_n(\mathbb{C^\mathbb{\infty}})$, and $G_n(\mathbb{C}^\mathbb{m})$ for some $m$?
This is true if $X$ is compact. It's not true for all paracompact spaces; I don't know if there's anything more you can say but for when $X$ is compact. (To see the second sentence, take $X = \Bbb{CP}^\infty$, $n=1$, and you get by looking at cohomology that every map $X \to \Bbb{CP}^k$ induces the zero map on cohomology; in particular there is no $k$ such that the identity map $X \to X$ factors through a map $\Bbb{CP}^\infty \to \Bbb{CP}^k$.)
When $X$ is compact, then note that every compact subset of $G_n(\Bbb C^\infty)$ is contained in one of the $G_n(\Bbb C^m)$s. So in particular the image of $X$ is, and your map (1) is surjective. But it's also injective, by the same logic applied to $X \times I$.
Now, because simplices and spheres are compact, similar logic implies $H_k(G_n(\Bbb C^\infty)) = \lim H_k(G_n(\Bbb C^m))$, and similarly with $\pi_k$ and $H^k$.
(I can't quite answer the de Rham question because I don't know what the de Rham cohomology of $G_n(\Bbb C^\infty)$ even means, given that it's not a finite-dimensional manifold.)