Avalanche looks like a French word built from the verb for "go down." The evidence is murkier than that.
Etymonline says the word entered English in 1763, for a mass of snow sliding down a mountain. Merriam-Webster dates the noun to 1744, so the exact first use is open. English took it from French avalanche, which Etymonline traces to the Romansh word avalantze, "descent."
Earlier, the word probably came from the Savoy dialect form lavantse and Provençal lavanca. Merriam-Webster likewise gives a French dialect form, lavantse or avalantse, as the source.
Behind the Alpine dialects, the trail goes cold. Etymonline says the word may trace to a pre-Latin Alpine language, possibly Ligurian.
The tidy story you will see elsewhere, that avalanche simply comes from Latin avalare or French avaler, "to descend," is therefore at best part of the picture. Etymonline says the form was probably reshaped under the influence of Old French avaler, but the word's root is Alpine.
Once in English the word moved around. It began with snow and later stretched to falling rock and landslides. A shortened form, valanche, was recorded by 1766, and avalanche as a verb is recorded from 1872.
Merriam-Webster records the verb earlier, in 1826 for the intransitive sense, against Etymonline's 1872, so the two dictionaries disagree on that date as well. Both treat it as an English extension of the noun.
The figurative use, an avalanche of mail or of criticism, rests on the same picture of something small starting to slide and then growing out of control.