In Liu's Algebraic Geometry and Arithmetic Curves, Proposition 1.2.6 states that given any short exact sequence $0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ with $M''$ flat, taking the tensor product of this with any module $N$ gives an exact sequence $0 \rightarrow M' \otimes N \rightarrow M \otimes N \rightarrow M'' \otimes N \rightarrow 0$.
Later in the book, in Exercise 5.2.8, page 193, he suggests using this Proposition to show that given a long exact sequence which is eventually $0$:
$$ 0 \rightarrow M_0 \rightarrow M_1 \rightarrow \cdots $$
of flat modules, taking the tensor product with an arbitrary module still gives an exact sequence
$$ 0 \rightarrow M_0 \otimes N \rightarrow M_1 \otimes N \rightarrow \cdots$$
How does this generalization follow? It appears that in the proof of the Proposition, he doesn't really use the surjectivity of $M \rightarrow M''$, but then again, all he proves is that $M' \otimes N \rightarrow M'' \otimes N$ is injective; the full exactness being something that always happens for tensor products of short exact sequences.
Best Answer
In fact, Liu says (on page 193, Exercise 2.8(b)) that the sequence is "zero from a finite rank on". Then you start the breaking from the rightmost side of the sequence: $0\to X_{n-1}\to M_{n-1}\to M_n\to 0$, and so on. You know that $M_{n-1}$ and $M_n$ are flat, so $X_{n-1}$ is also flat, and so on.