What attempts did the authorities in Moscow try to deescalate the situation? How did both SSRs acquire military equipment and men, were they defecting Soviet troops with ethnic loyalties?
The actual war did not begin in 1988. What happened was in February that year the Nagorno-Karabakh National Assembly voted to secede from the Azerbaijani SSR and join the Armenian SSR; this was not accepted by Azerbaijan, and the result was severely strained ethnic tensions amongst the populations of all three regions, with mass protests in both Yerevan and Stepanakert (the capitals of Armenia and NK, respectively).
Seeing how similar ethnic tensions had led to outright massacres of Azerbaijanis and Armenians in 1905-1907, the Soviet authorities moved to try and quiet the situation, sending in armed forces (I can't fully recall if it was just interior ministry soldiers or outright army troops). This didn't alleviate the issue, as events like the Sumgait pogrom (which saw officially at least 26 ethnic Armenians killed outside of Baku, the Azerbaijani capital) showed. Tensions stayed high though, and with the breakup of the Soviet Union in 1991 and the subsequent independence of both Armenia and Azerbaijan, outright war broke out in 1992.
As for the acquisition of military equipment, there were units based in each republic comprised of the titular nationality that went over to their respective side when the USSR ended, and that comprised the basis of the Armenian and Azerbaijani armed forces.
For further reading, there is no better source on the conflict that Thomas de Waal's Black Garden: Armenia and Azerbaijan Through Peace and War (2003; 2013). It is the most balanced account of the history of the conflict and doesn't take sides, and has become the standard account of the Nagorno-Karabakh, and indeed Armenian-Azerbaijani relations. It's a real easy read as well, and de Waal is still posting updates on the region through the Carnegie Endowment for International Peace.