On modular computation of Gröbner bases with integer coefficients

Let $I_1\subset I_2\subset\dots$ be an increasing sequence of ideals of the ring $\mathbb Z[X]$, $X=(x_1,\dots,x_n)$ and let $I$ be their union. We propose an algorithm to compute the Gröbner base of $I$ under the assumption that the Gröbner bases of the ideal $\mathbb QI$ of the ring $\mathbb Q[X]$ and the the ideals $I\otimes(\mathbb Z/m\mathbb Z)$ of the rings $(\mathbb Z/m\mathbb Z)[X]$ are known.
Such an algorithmic problem arises, for example, in the construction of Markov and semi-Markov traces on cubic Hecke algebras.