The barbarian threw the druid up into the air and the druid had animal shapes held as an action with the trigger "once in range" ;)

Yes, this is what I live for.

Some of the conjure spells don't give the stat blocks in the text of the spell, they list and link to the creatures you can summon as it wouldn't make sense to list them all with the spell. There are usually more creatures than what's listed, it depends on what creatures were around when it wasprinted, but it does give you stat blocks if it expects you to have a creature

Bag of tricks is particularly bad example as it only summons those monsters which are listed, which is more restrictive than most conjuration spells, and is an item.

Evard's Black Tentacles and Cloudkill also summon a persisting damaging effect like insect plague does, but I still wouldn't let you animal shape those.

Unseen Servant doesn't even summon a full creature but it does list some stats and I'd probably rule you could animal shape as it at least has HP and AC to go off of.

Where in insect plague does it say it summons a swarm of insects? It doesn't. If it did, it would that it conjures a swarm of insects and hyperlink the monster listing. It repeatedly referrs to the magical effect as either the sphere or the area.

Insects plagues effects also don't line up the stats of swarm of insects,

4d10 damage, 20ft radius sphere, con save for half damage, cannot move

vs

medium sized swarm of tiny creatures, melee attack +3 to hit, 4d4 damage or 2d4 when >= half hp, can move 20ft

This actually brings up a good point...

How many insects are in a swarm? The stat block doesn't actually list an amount, so how many (Large or smaller creatures) are going to appear?

And aren't they going to to appear in the same space, since the swarm is copnglomerate medium creation, that would be several large creatures in one 5x5 space, would they even survive? Or are you going to be raining chunks of horse all over the battlefield?

Even if you said a 20ft radius sphere, there could potentially thousands of locusts in there. If you rule that they all move to an unoccupied space, your most likely going to cover your entire map with horses, horses fall and everyone dies.

On the range, I was just being precise about the real distance, but now that you mention, the druid would have to be within 30 feet to start the combo. Unlike actual summon spells, you have no way of communicating with or commanding the swarm of insects to fall. It can't speak, with no listed languages and too low intelligence to learn one. Insect Plague is a magical effect with no senses whatsoever, and can't move.

This loops back around to the whole "willing" part of animal shapes. Swarm of insects has no way of being willing or even communicating it's willingness to be transformed. You could try charming the swarm of insects, except that they are immune to the charmed condition.

They are also immune to being knocked prone, so you can't use that to make them fall as you might for another flying creature. Really they are immune to most other conditions, I can't see one that would let you circumvent the inability to be willing or just mechanically make them fall.

There is no hard rule for one creature falling onto a other. There is however an optional rule in Tashas that states:

If a creature falls into the space of a second creature and neither of them is Tiny, the second creature must succeed on a DC 15 Dexterity saving throw or be impacted by the falling creature, and any damage resulting from the fall is divided evenly between them. The impacted creature is also knocked prone, unless it is two or more sizes larger than the falling creature.

Without this optional rule, we have no RAW or RAI way of determining the damage this would even cause.

Weight and size do not matter for fall damage, so you would actually be better off using medium creatures as you could hit large or larger creatures multiple times. Elephants are still really funny, although I would have used cows.

The worst part about all this combo that clearly does not work is that it uses a 5th and 8th level spell slot from two different PCs, or one over multiple turns, to do something that you can completely RAW using one 3rd spell slot to cast.

Conjure Animals lets you summon eight CR1/4 creatures, in an unoccupied space with a range of 60ft. Normal falling damage is 1d6 per 10 feet fallen, round down. So one conjured creature could be 5d6 damage. If you have a huge or gargantuan target, or multiple targets, this could be as high as 40d6. That would be enough to kill any gargantuan creature below CR11 assuming average HP.

Sorry if this is incomprehensible or too long, I wrote this over a couple hours in between errands and making dinner, but I did enjoy going over everything that goes into this.