Even though the bistellar moves are motivated by concordance and the bistellar moves are not, it is not hard to make concordance triangulations for stellar moves as well.

Even though the bistellar moves are motivated by concordance and the stellar moves are not, it is not hard to make concordance triangulations for stellar moves as well.

There are then two levels to your question. A triangulation $T$ of $X$ induces a piecewise linear structure $\mathcal{P}$. You could ask whether the PL structures $\mathcal{P}$ and $\mathcal{P}'$ are isotopic or concordant, without worrying about the original triangulations. For simplicity suppose that $X$ is a closed manifold. Then at least in dimension $n \ge 5$, Kirby-Siebenmann theory says that the set of PL structures on $X$ up to isotopy are an affine space (or torsor) of $H^3(X,\mathbb{Z}/2)$. I think that the concordance answer is the same, because the Kirby-Siebenmann invariant comes from the stable classifying space of the germ-theoretic tangent bundle of $X$. In other words, two PL structures give you a map from $X$ to $\text{TOP}(n)/\text{PL}(n)$, which means germs of homeomorphisms divided by germs of PL homeomorphisms. Stabilization in this case means replacing $n$ by $\infty$ by adding extra factors of $\mathbb{R}$. If $n = 4$, then up to isotopy there are lots of PL structures on many 4-manifolds, as established by gauge theory. But I think that the concordance answer is once again Kirby-Siebenmann. (I learned about this stuff in a seminar given by Rob Kirby — I hope that I remembered it correctly! You can also try the reference by Kirby and Siebenmann, although it is not all that easy to read.)

The other half of the question is to give $X$ a distinguished PL structure $\mathcal{P}$, and to look at triangulations $T$ and $T'$ that are both PL with respect to $\mathcal{P}$. In this case there are two good sets of moves to convert $T$ to $T'$. First, you can use stellar moves and their inverses. A stellar move consists of adding a vertex $v$ to the interior of a simplex $\Delta$ (of some dimension) and supporting structure to turn the star of $\Delta$ into the star of $v$. The theorem that these moves suffice is called the stellar subdivision theorem. (The theorem is due to Alexander and Newman and it is explained pretty well in the book by Rourke and Sanderson.) The other set of moves are specific to manifolds and they are the bistellar moves or Pachner moves. One definition is that a bistellar move is a stellar move that adds a vertex $v$ plus a different inverse stellar move that removes $v$ (hence the name). But a clearer definition is that in dimension $n$, a bistellar move replaces $k$ simplices by $n-k+1$ simplices in the minimal way, given by a local $n+1$ concordance that consists of attaching a single $n+1$-dimensional simplex. The theorem that these moves work is due to Pachner. Pachner's moves in particular give you a shellable triangulation of $X \times [0,1]$.

There are then two levels to your question. A triangulation $T$ of $X$ induces a piecewise linear structure $\mathcal{P}$. You could ask whether the PL structures $\mathcal{P}$ and $\mathcal{P}'$ are isotopic or concordant, without worrying about the original triangulations. For simplicity suppose that $X$ is a closed manifold. Then at least in dimension $n \ge 5$, Kirby-Siebenmann theory says that the set of PL structures on $X$ up to isotopy are an affine space (or torsor) of $H^3(X,\mathbb{Z}/2)$. I think that the concordance answer is the same, because the Kirby-Siebenmann invariant comes from the stable germ-theoretic tangent bundle of $X$. In other words, two PL structures give you two different sections of a bundle on $X$ whose fiber is $\text{TOP}(n)/\text{PL}(n)$, the group of germs of homeomorphisms of $\mathbb{R}^n$ divided by the subgroup of germs of PL homeomorphisms. Stabilization in this case means replacing $n$ by $\infty$ by adding extra factors of $\mathbb{R}$. If $n = 4$, then up to isotopy there are lots of PL structures on many 4-manifolds, as established by gauge theory. But I think that the concordance answer is once again Kirby-Siebenmann. (I learned about this stuff in a seminar given by Rob Kirby — I hope that I remembered it correctly! You can also try the reference by Kirby and Siebenmann, although it is not very easy to read.)

The other half of the question is to give $X$ a distinguished PL structure $\mathcal{P}$, and to look at triangulations $T$ and $T'$ that are both PL with respect to $\mathcal{P}$. In this case there are two good sets of moves to convert $T$ to $T'$. First, you can use stellar moves and their inverses. A stellar move consists of adding a vertex $v$ to the interior of a simplex $\Delta$ (of some dimension) and supporting structure to turn the star of $\Delta$ into the star of $v$. The theorem that these moves suffice is called the stellar subdivision theorem. (The theorem is due to Alexander and Newman and it is explained pretty well in the book by Rourke and Sanderson.) The other set of moves are specific to manifolds and they are the bistellar moves or Pachner moves. One definition is that a bistellar move is a stellar move that adds a vertex $v$ plus a different inverse stellar move that removes $v$ (hence the name). But a clearer definition is that in dimension $n$, a bistellar move replaces $k$ simplices by $n-k+1$ simplices in the minimal way, given by a local $n+1$ concordance that consists of attaching a single $n+1$-dimensional simplex. The theorem that these moves work is due to Pachner. Pachner's moves in particular give you a shellable triangulation of $X \times [0,1]$.

Even though the bistellar moves are motivated by concordance and the bistellar moves are not, it is not hard to make concordance triangulations for stellar moves as well.