When do Schubert polynomial products stabilize?

The "back-stabilization number" for products of Schubert polynomials is the distance the corresponding permutations must be shifted before the structure constants stabilize. We give an explicit formula for this number and thereby prove a conjecture of N. Li in a strengthened form. This leads to an additional result: a formula for the smallest $n$ such that a given Schubert product expands completely over $S_n$. Our method is to explore back-stable fundamental slide polynomials and their products combinatorially, in the context of their associated words. We use three main tools: (i) an algebra consisting of "colored words", with a modified shuffle product, and which contains the rings of back (quasi)symmetric functions as subquotients; (ii) the combinatorics of increasing suffixes of reduced words; and (iii) the lift of differential operators to the space of colored words.