They are still very young... night-time "dryness" is something that is not fully attained until even 7 years old. It is biological based development. So, they could still have night-time diapers on, for bed. Even Kindergarten kids still wear night-time diapers at night, for example. Normal.
You have just started potty training. It takes time. They will not attain peeing/pooping/AND night-time dryness and going to the toilet at night themselves all at the SAME time. It occurs in stages. Gradually.
So, you are still at the beginning of potty training. No rush. Yes, it is a slow process.
Them switching to a toddler bed, should have no bearing on their night-time dryness expectations or ability. If they wet at night, which is normal still... then just use waterproof bed-pads under them, to make clean up easier, if their diapers leak. That is what I do.
Mostly, do not "expect" a child, at this age... to just get up at night and go to the bathroom at will, perfectly and to not have accidents. Again, it takes even up until 7 years old, for night-time dryness to occur. But in the mean time their bodies and their ability will mature, and one day they will "master" it. So until then.... take the night-time potty expectations separately. They are still trying to daytime potty train and have not mastered that portion of it yet. So that is only the 1st stage of potty training.
Potty training occurs in 3 stages:
1) daytime peeing in toilet and ability
2) pooping ability in toilet and mastering it
3) night-time dryness and mastering it.
But keep in mind accidents will still occur. Normal.
My daughter was still wearing a night-time diaper until 5 years old. MANY kids do, even at this age. Per our Pediatrician, it is NORMAL.
Also as a side tip: a "car potty" is real useful for times a child needs to pee but you are in the car or on the road and no bathroom in in sight or you are stuck in traffic. We use one and it is a lifesaver. Here is a link:
http://www.amazon.com/s/ref=nb_sb_noss?url=search-alias%3...
All the best,
Susan