One notion of framed cobordism is that $\Omega_{fr}^n(X)$ is a certain equivalence class of $n$-manifolds $M^n$ in $X$ with a trivialization of the normal bundle $N M^n\subseteq TX$. For compact manifolds, there is the correspondence $\Omega^n_{fr}(X)\simeq [X,S^{m-n}]$.
Another definition I have seen is free of the ambiend manifold $X$: a framing on $M^n$ is the trivialization of the stable tangent bundle $\tau_M\oplus\epsilon_M^1$ and a framed cobordism class $\Omega_{fr}^n$ is defined without $X$.
Are these two different mathematical objects, or is there some simple connection, such as $\Omega_{fr}^n\simeq \lim_{N\to\infty} \Omega_{fr}^n(S^N)$ in "some sense"? If there is a clear connection, why?
According to the definition I have seen $\Omega^n_{fr}(X)$ is the set of cobordism classes of continuous functions $f\colon M^n\to X$. The map $f_1\colon M\to X$ is said to be cobordant to $f_2\colon N\to X$ if there is a cobounding $(n+1)$-manifold $W$ so that the maps $f_1$ and $f_2$ can be extended over all of $W$ to a map to $X$. I'm spelling this out because your post makes it sound like these manifolds have to be embedded in $X$. Also I am suppressing framing information. Now the bordism groups $\Omega^n_{fr}$ which are defined without reference to an ambient manifold are actually just the bordism groups of a point $\Omega^n_{fr}=\Omega^n_{fr}(*)$. There is an obvious homomorphism $\Omega^n_{fr}(X)\to \Omega^n_{fr}(*)$.