Touring with kids - tag along and kids trailer behind same bike

Sounds like a bad idea. The biggest concern would be how the setup would act under hard braking, especially when going down an incline. There will a lot of mass to stop which means your stopping distance will be very long. Additionally the trailer and tag-along chain will be unstable under braking and may jack-knife.

Update:

If you are going to try this in a controlled fashion as Criggie suggests in his answer, Make sure your brakes are in good shape and properly adjusted first. Having a pro mechanic at a bike shop inspect and set them up would be a good idea.

Those of us who tow bike trailers have generally encountered a situation where some other road user underestimates your total length.

If your road-train bike was three units long there's more opportunity for someone else to screw-up and clip the end of your setup.

An off-road path or a park would be a much safer way to try this - if you intend to ride this on the road then I strongly recommend a flag on a pole for each unit. Perhaps even flag bunting between the poles (ie a string with little flags on it)

If you already own the three parts, there's no harm in trying it. Start by rigging the train empty and see how it handles in a safe location. For a better test, add some non-live ballast to the trailer (easy) and perhaps the tagalong (harder to do)

OPINION: It looks quite fun, but I'd not ride that rig on a public road.

I tow a 18 foot canoe with my bike, which is much longer and heavier then your setup. As Criggie says use lots of flags. Add a dash of common sense and give it a try.

Related post How to tow a canoe with a bicycle?

I towed this setup. The only difference, second trailer was smaller and used only for storage. We put there some supplies and small kids bike, so when we get to the park, child can drive on her own.
On straight roads no problem. Going up hill especially in windy weather can require some sweating. Maneuvering on streets taking turns, coming to press traffic light button (If there is no bike line, I'm using sidewalk) can be tricky and needs some practice.