In European colonial "small wars" the Europeans fought everyone from cultures which had been civilized while Europeans were still barbarians to stone age tribes. So naturally the use of artillery by the opponents of the Europeans was not nearly as consistent as the use of artillery by the Europeans, which itself was highly variable.
In "small wars" with relatively small forces engaged each combatant, each musket, each rifle, and each artillery piece was much more important than it would be in a larger combat. So there were battles where the presence of even a single artillery piece gave a decisive advantage to the Europeans.
For example, at the First Battle of Adobe Walls, November 25, 1864, many credit the two artillery pieces with holding off the Southern Plains warriors and preventing the massacre of Carson's entire command.
Thus the victories of the non Europeans over Europeans were often the victories of forces with fewer or no artillery over European forces with more artillery. There were exceptions, like the Battle of Pollilur in 1780, where the Mysore forces won with superior rocket artillery, and the Battle of Adwa 1896 where the Ethiopians with 42 superior artillery pieces defeated the Italians with 56 older cannons, for example.
Some examples of forces with no artillery defeating forces with artillery include:
1) Battle of the Monongahela 9 July 1755 General Braddock's force of over 2,000 men with 10 cannons was crushed by a smaller French and Indian force with no artillery.
2) St. Clair's Defeat, November 4, 1791. The Western Confederation defeats the US forces killing over 800 Americans. The surviving artillerymen spike their cannons and flee.
3) The Dade Massacre December 28, 1835. 110 US soldiers with one cannon are ambushed by Seminoles. 3 soldiers escape but one dies of his wounds.
4) The retreat from Kabul January 6-13, 1842. 4,500 British and Indian soldiers and 14,000 civilians, with 6 artillery pieces, agree to evacuate Kabul, Despite the truce, thousands of tribesmen with muskets constantly ambush and attack the retreating forces. There are not many more survivors than the single man told of by legend.
5) The Grattan Massacre August 19, 1854, the start of the First Sioux War. 30 soldiers and one civilian with 2 cannons are slaughtered. The only soldier who escapes dies of his wounds.
6) The Battle of Pine Creek May 17, 1858. Edward Steptoe and 159 soldiers are trapped on a hillside. The Indians withdraw for the night and the soldiers know they will be slaughtered the next day. They abandon their supplies and two mountain howitzers and sneak away to safety in the night.
7) The Battle of the Big Hole, August 9-10, 1877. The Nez Perce camp is surprised by soldiers under Colonel Gibbon but the Nez Perce rally and keep the soldiers pinned down for two days. The crew of Gibbon's howizter is killed or driven off.
8) The Battle of Isandhlwana 22 January 1879. A large British column is almost totally slaughtered and two mountain guns and a rocket battery are lost.
9) The battle of Laing's Nek 28 January 1881. A British force with 6 cannons is repulsed by Boers with no canons.
10) The Shangani Patrol of 34 men in pursuit of the defeated Matabele King Lobengula was wiped out on 4 December 1893 while other Matabele warriors on the other side of the Shangani River attacked the main column under Major Forbes. Despite having Maxim machine guns, Forbes was unable to cross the river and retreated after the fight, being attacked by the Matabele several times in the 2 week journey back to Bulawayo, with many in Forbes' command believing they narrowly escaped being massacred.
11) The Battle of Togbao October 10, 1898. Lt. Bretonnet with 50 french and colonial troops and 400 Baguirmian allies was defeated by Rabah Zubayr. Rabah's men slaughtered most of Bretonnet's men and captured his 3 canons.
12) The Battle of El Herri 13 November 1914. A strong French force was crushed, with 623 officers and men killed and 176 wounded, by Zaian tribesmen despite the French having two batteries of cannons and 4 machine guns.
13) The Battle of Annual July 22, 1921 was a terrible Spanish defeat by Rif tribesmen, followed by many other Spanish defeats up to August 9. Spain officially lost 13,192 soldiers killed July 22 to August 9, with 171 wounded, but some sources claim the casualties were much higher. The previously poorly armed Rifs captured thousands of rifles, and 60 machine guns and 100 cannons according to one source, or 400 machine guns and 120 to 150 cannons according to another source.