  In arXiv:1905.07734 we presented a construction that is an analogue of
Pontryagin's for proper maps in stable dimensions. This gives a bijection
between the cobordism set of framed embedded compact submanifolds in
$W\times\mathbb{R}^n$ for a given manifold $W$ and a large enough number $n$,
and the homotopy classes of proper maps from $W\times\mathbb{R}^n$ to
$\mathbb{R}^{k+n}$. In the present paper we generalise this result in a similar
way as Thom's construction generalises Pontryagin's. In other words, we present
a bijection between the cobordism set of submanifolds embedded in
$W\times\mathbb{R}^n$ with normal bundles induced from a given bundle
$\xi\oplus\varepsilon^n$, and the homotopy classes of proper maps from
$W\times\mathbb{R}^n$ to a space $U(\xi\oplus\varepsilon^n)$ that depends on
the given bundle. An important difference between Thom's construction and ours
is that we also consider cobordisms of non-compact manifolds after indroducing
a suitable notion of cobordism relation for these.
