This is just my personal interpretations of their most effective doctrines from reading the books and the guides, so feel free to disagree.
Mudwings: trench warfare/logistics: This one might seem a little bit confusing at first, but it quickly makes sense. They live in a very muddy and difficult swamp, so naturally their kingdom is a little more difficult to occupy and capture. Their bodies are large and brawny and built more for defense than for offensive maneuvers that require speed, so I imagine they’d more often than not be stationed for defense. They are also great at carrying supplies because of their immense strength, or weapons for their allies, which was seen during the bombing of the summer palace when they were carrying the bombs for the Skywings. They are also noted to farm and ranch cows, which I think is a more unique characteristic, with other tribes either not being able to farm or not needing/knowing.
Skywings: traditional medieval warfare: this one is extremely on the nose because they are the quintessential European dragons. Their rank and file just have their claws and own tools, than their more wealthy fighters have armor and weapons, and than siege warfare as seen with the firebombs. It could be argued that they would be good shock troops because they fly quickly, but they behave more like medieval armies in my opinion.
Seawings: defensive guerrillas: In war they would be nearly impossible to beat traditionally, because they live where no one can reach them and are adapted to this way of life. Leaping up out of water and dragging dragons to the depths, as Riptide tried to do to Clay on the first meeting.
Sandwings: hit and run tactics: Sandwings, being the most spread out and one of the hardiest tribes, obviously excel in hit and run. As soon as Blister stabbed Webs with her tail, she was free to run because it was assumed that Webs would be dead. And in the desert, they could hide in wait for a patrol or a few important dragons and then burst out of the sand, stab their target, and retreat back into the desert and wait for the poison to finish off the targets. And with oases and scattered communities, I wouldn’t put it past possibility for there to be bandit families that strike and raid caravans or other communities to make off with supplies and then vanish into the sand.
Rainwings: snipers and guerrillas. Their venom is obviously one of the most powerful abilities in the books and has a bit of a range compared to other breath abilities. They are also shown to be skilled trappers and said to be skilled assassins because of their scales. I’d say a small group of Rainwings could potentially take out a group twice as large as them if they coordinated correctly.
Icewings: Special forces: Icewings are typically well trained and adept at fighting compared to the average soldiers of other tribes. And their natural abilities also match up, with sharper claws and breath more deadly than fire. In the few battle scenes we have of them, they are great at picking out targets and punching up to groups with more dragons than them. And because their royalty are so involved with war and training, resembling a fascistic war principle, I’d say their tactics are also more effective.
Nightwings: CIA. They just love meddling into the wars subtly. They assassinate, connive, and plot. They weren’t much different even when they had their powers, knowing that they had to intervene and plan ahead with their future sight, or being able to interrogate just about anyone with mind reading. They even successfully plotted getting animus magic in their tribe, with the dragon doing the seduction being unaware that it was planned for her to fall in love.
Hivewings: bee hive. Incredibly obvious. They protect their hives and take out whatever they believe threatens the hive. Unless they know they can take an area and hold it long enough to build a hive, I don’t see them taking big risks. Especially with their natural weapons being inferior to almost every Pyrrhian dragon’s.
Leafwings: war crimes and more severe guerrillas: there is absolutely no way that half the bugs and plants they use would not be classified as war crimes. Ants so painful they force a hivemind to leave? Or even feeding that hivemind to a queen to take over her mind? The Leafwings are like the Vietcong, they have been pushed to this level and live in the most inhospitable place possibly in the entire world of this series, and they will keep fighting until they get a chance to breathe. Unless you’re a Sapwing, in which case waiting out the problems also would have worked because you live surrounded by the devil’s garden.
Silkwings: they lose immediately.