typically cannons, spells from lore of death, charging them with a unit with several ranks - thus ensuring you remain steadfast throughout several rounds of combat & reduce your risk of breaking - most armies have one or two units of cheap, throw-away core units that are specificalyl designed for this purpose.
alternatively you could get your own big & nasty unit to counter-punch with.
With my Vamps I like to use ghouls poison and augment spells. Get in with an ASF or initiative boost, stack it with re-rolls if you can, then wail away with a bucket load of dice.
Some sixes usually pop up, and with the re-rolls its usually enough to seriously hurt the bigger beasties. Start with a big enough unit and you should be able to withstand a round or two of that till it drops - just try and avoid the riders issuing challenges by refusing them. Better to lose by a point or two more than to lose the character permanantly with no combat res to add to the undead pile. Then when the beastie is dead, take out the character the same way.
Its the long view, but it usually works, and can be seriously cost effective.
As someone who plays 4 armies without a single cannon between them, I like to throw lots of dice at them. The fact that even T8 monsters can go down to S2 hits is why they're finally a balanced element in 8th edition
Turn it into swiss cheese with cannons; force characteristic tests on its worse stats (often I, magic time); use poisoned stuff; shoot it with everything you've got; get a monster of your own; smack it repeatedly with high-S weaponry (or auto-wounding ones); or simply disable it by confuddling it.
Of course, hitting it with a horde in the side should break it. 50 skavenslaves is really cheap, and should force 10 combat res points from the front and charging, 11 on the side and 12 from the back. Then they attack. Short of dragons and breath weapons (which they can only use once), you shouldn't lose.
Everything has access to poison, characteristic test magic, cannons or counter-gribblies.
my experiance of trying use of wyvern or hippigriff mounts for chaos loards has met with mixed success. attacking the dragon not to rider seems to be the way to go - even with high T it is much easier to hit against the dragon's WS and then damage than the other way around.
Dragons and wyveryns and similar monsters are more rare in 8th edition. The greater daemons of daemons of chaos are also more rare now. The first reason is because cannons and certain other war machines (doom divers, even direct hits by stone throwers) can kill them too easily and poisoned shooting and attacks and certain magic can also take them down too easily. The second reason is that, with the new steadfast and ste up rules, certain units can tie them up and wear them down with wounds over time since a roll of 6 to wound autowounds and most monsters do not have a great armour and ward save (regen on hellpit aboms and hydras are exceptions for those units). Third, there are a lot of characters and models with high strength that can deal with monsters (including flaming attacks) that can hold up against a monster (warriors of chaos lord, a dreadlord, an oldblood with great weapon, ironguts, GW special units, etc.). Finally, large monsters eat up a lot of the points allotments either in the character categories (when mounted by a character) or in the rare slot. Many armies that ran characters on monsters in 7th edition now prefer a fighting lord and a lord mage to fill out their lord points allotment.
Whether to attack the mount or the rider depends on the situation and relative saves. A high elf dragon lord with re-rollable high armour save and 4+ ward is harder to wound and kill than a T6 dragon with minimal save, especially if you have S5 and S6 attacks. I've often won combat with static CR (banners, ranks) plus wounds on the monstrous mount sufficient to force a break test and break the lord from combat.
Lizardmen get a bit lucky with their monsters. The Carnosaur not only has 4+ Scaly Skin (I think) but it does NOT have the Large Target Rule, meaning if you put an Oldblood on it, you don't suffer from the Cannon Snipe as much. Not to mention it's sweet stat line and rules.... Luckily for the opponent it has crap WS.
If the monster has low Initiative, use Pit of Shades on them, instant kill if it fails the test (or is it wound?) can't remember. Use weak units, that are cheap and expendable, if they have poison rule, even better. It's going to require 6's anyway.... So just throw waves at it. Or, if you like cinematic type games, charge your own big monster or IC at them.
Purple Sun, Pit of Shades, Final Transmutation, Dreaded Thirteenth (not sure if this is viable), pretty much any of the 'I-eat-expensive-shit' power-spells. Debuffs also work, so stacking Plague of Rust or something is a good plan. If you're blessed with a good leadership then a block of dudes with Mindrazor are also quite tasty.
Dreaded Thirteenth only works on infantry. Final transmutation only kills on a roll of 6. Purple Sun and Pit of Shades work on low to medium initiative, particularly effective against stegadons, hydras, Ogre Kingdoms monsters, and similar low I models but some monsters have greater I and are less likely to be killed unless on casts miasma first to lower I and then pit of shades.
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