For the purposes of this question, let's say that a _blurring_
of a smooth triangulation T of a smooth manifold X
is a smooth homotopy h: [0,1]×X→X such that h_0=id_X,
h_t maps each simplex of T to itself,
and h_1 maps some open neighborhood of each simplex σ of T to σ.

As an example, consider the n-sphere S^n={x∈**R**^{n+1}:‖x‖=1} triangulated
into 2^{n+1} simplices by the coordinate hyperplanes x_i=0.
A blurring for this triangulation can be constructed as h_t(x)=proj(C_t(x)),
where proj(x)=x/‖x‖,  C_t(x_0,…,x_n)=(c_t(x_0),…,c_t(x_n)),
c_t(r)=(1−t)r+t·b(r), and b is a smooth bump function such that
b=id on (−∞,−ϵ]∪[ϵ,∞) and b=0 on [−ϵ/2,ϵ/2], where ϵ<1/(n+1).

**Does every smooth triangulation of a smooth manifold admit a blurring?**

Madsen and Weiss in §A.1 of their paper on Mumford's conjecture
incorporate the above notion of blurring into their notion
of an _extended triangulation_ (which also involves a total ordering on vertices of T,
an extension of simplices of T to extended simplices in a compatible way,
and a requirement that h_t preserves extended simplices).
They assert without proof that any smooth triangulation can be extended
to an extended triangulation, which in particular would imply a positive
answer to the above question.
This assertion is an important step in their proof of Mumford's conjecture.

Some obvious approaches to constructing blurrings fail for various reasons.

For example, one can construct a family of smooth maps b_n: Δ^n→Δ^n,
functorial in the simplex n, such that b_n maps an open neighborhood of ∂Δ^n to ∂Δ^n.
Using a linear interpolation between b_n and the identity map on Δ^n,
one can construct a _continuous_ homotopy h that satisfies all the properties of a blurring except smoothness.
However, such an h is not smooth on simplices of codimension 1 and higher.

Another approach tries to construct h by induction on the skeleta of T.
For example, given a map h: [0,1]×T_n→T_n defined on the n-skeleton T_n of T satisfying the above properties,
one could try to extend it to the (n+1)-skeleton T_{n+1} of T using (very roughly) the following three steps:
(a) for each n-simplex σ of T construct an extension of h to some open neighborhood of the interior of σ inside T_{n+1};
(b) assemble all maps constructed in part (a) into a single extension of h to some open neighborhood of T_n inside T_{n+1};
(c) extend the map in (b) from the open neighborhood of T_n to T_{n+1}.
Even if one succeeds at (a) and (c), it is not at all clear to me what to do about (b), because there is no reason why all the individual extensions should be compatible near the (n−1)-simplices of T.

I'm also open to imposing additional conditions on smooth triangulations,
as long as one can show that any smooth manifold admits a smooth triangulation with such additional properties.