Probably obvious, but I have no babies of my own. I’m guessing there aren’t any full size portable cribs. Do they just sleep in the little baby carrier? If so, where do you put it? On a dresser, in bed with you? Or in the floor? Or on a blanket on the floor surrounded by one of those baby cages? Looking for a general answer, such as you would find in a parenting advice book, but anecdotes are OK too.
There are full-size portable cribs. They fold up.
In the car they sleep in their car seats. And in a motel room most people would bring their own pack and play’s. But if this all escapes you maybe think of not having any yet. 
Just tuck them into bed with you.
There are a variety of portable cribs (in the UK they’re called travel cots), ranging from a very light popup box which serves till they’re about 4 months old, to a big solid thing with a metal frame which they can use till they are about 2 years old. The travel cot we bought served as a playpen as well (somewhere enclosed to put the baby to play for short periods), so that will give you an idea of the size. It also had a travel changing table to put across the top. As it happened, it weighed more than a minibus so it was impossible to travel with it anywhere at all, so we’ve only ever used it when friends brought their children to us.
It also depends where you’re travelling - most hotels will provide some sort of cot, although you often pay for it, and if you’re visiting another parent, they’ll have something left over from their own children often. Worst case, we’ve made a bed out of pillows and blankets and popped her in her grobag (baby sleeping bag), but then sleeping has never been a problem with my daughter.
A more difficult problem, in my experience, has been working out a way to feed newly weaned babies whilst travelling - a travel high chair is more unusual.
Babies can sleep almost anywhere, right? Couldn’t you just stick them on the floor on a blanket or something? Or in bed with you?
The smallest ones are like two feet long and flat. They actually seem pretty portable.
I would imagine, but are there smothering issues?
Generally not. Unless you are drunk, drugged or grossly obese, smothering is much less likely than, say, cot death.
Considering how rock-hard most portacot mattresses are, you almost might as well be sticking them straight on the floor. The reason for the portacot for tiny babies is simple - to prevent them from being stepped on by careless Big People. The reason for older, more mobile babies - to make sure they stay where you put 'em.
*How do you travel overnight with babies? *
Based on extensive observations on airlines over the years, I believe the answer for 90% of the population is “Extremely noisily”. 
The travel cot Middlebro and his wife have doubles as a playpen. Runner-up for “greatest invention since sliced bread” when you have a kid who’s just barely old enough to crawl around (and do they crawl fast!) but not enough to understand danger.
Their stroller has three “baskets”: pram, big baby (aka “the egg”) and toddler. Both the pram and baby baskets are carseat-rated and can also double as cradles. The toddler basket is soft and not rated as a carseat.
Couldn’t they fall out of the bed?
If you have two people, the baby sleeps in between. If you have one person, you stuff a couple pillows under the covers on the other side of the bed to fence them in.
And falling is only an issue once they’ve learnt to roll (at about 4 months iirc) anyway.
We used to have a corral with a mat. Worked fine.
In hotels, you can always get a rollaway crib.
In addition to what everyone has suggested, there are companies that rent baby equipment for temporary use while on a trip.
I’ve used babysaway.com. They are a little on the expensive side, but very good. They have everything you might need, they’ll deliver to your temporary address and pick up when you leave. I have no connection to them except that I’ve used them a few times and been satisfied.
The babies? 
We did a bit of everything. Got cribs from the hotel service (no charge, brought a small portable crib/play pen and let the baby sleep with us. My kids generally slept with us until they were about 6 mos old, so that was the most common thing we did. Having the crib was useful for corralling the little guys while in the room, though.
If the bed is in the middle of the room with no wall on either side, do you just put pillows on both sides?
No, you would sleep on one side of them - the pillows are for the other side, if you don’t have another adult in bed.
We’ve done just about everything, depending on where we’re going, the mode of transportation and how much gear we wanted to lug along.
At first, it was super easy, as she slept in an Amby Motion Bed. It’s like a free standing baby hammock, and it breaks down and goes in its own tote bag. The whole thing when packed was a bit longer than a quad chair, and about four times the weight, but fairly easy to carry via the shoulder strap. It just went everywhere with us, whether it was visiting or camping or in a hotel. It was great, because it was her own bed from home, and familiar and comfortable.
Once they can sit up, they’ve outgrown that wonderful contraption, though. Then it was a mishmash of sleeping on the big bed with strategically placed pillows (a baby old enough to sit up on her own has the neurological maturity to wake up and move if her face gets smooshed on a pillow; only wee ones suffocate whilst sleeping), sleeping on a couch turned so the seat was up against the wall (with the arms and back making a “cage”), sleeping on the floor on top of a couple three blankets…whatever, really.
We never needed it when out, but when my son was born, his nursery wasn’t yet ready. He spent his first two weeks sleeping in a dresser drawer. Hey, I took the socks out first! I even took the drawer out of the dresser and placed it on the floor. Lined it with a few folded receiving blankets and he slept like a…well, you know. 