I spend a lot of time feeding preemies -- and teaching their parents to feed them. Often, the parents have the mistaken notion that practice makes perfect. For the adults involved, that may be true. For the babies, not so much.
There is a phenomenon among near-term preemies which demonstrates the issues pretty well for that group. The baby will poke along, choking and dribbling - sometimes failing to breathe as well. One day the baby just gets things together. It's almost magical - as if someone has flipped a switch. I've explained this to countless parents and recently a dad summed it up much better than I could. He said, "I see, it's like walking! He won't be able to do it until he can." The baby's mom was not exactly thrilled with any of us (dad, me, or baby), but she did seem to understand. Good thing. For that particular baby, it was weeks before he managed to pull his act together and nipple all his feedings.
Feeding preemies is much more complicated than feeding healthy term kids. With a term baby, you may have to help them figure out how to latch at the breast (or more rarely on an artificial nipple). Beyond that, the whole thing is pretty much reflex. Preemies have a pretty good suck reflex by 34-35 weeks gestation, but many don't get that swallow part. If they get the suck-swallow, they may not manage to coordinate breathing along with it. Eventually most of them will be able to manage, but it can be very frustrating for the parents while they're reaching that point.
Several things can help. First, don't cold stress the baby. They need to be warm - this often means spending time in an incubator, or at the least being dressed warmly and double-wrapped. Sometimes parents need a lot of education to understand that their preemie simply won't tolerate being unwrapped and passed around the group, especially prior to feedings. If the baby is using excess calories to keep warm, he won't have the energy to nipple or breast feed.
Preemies may not feed well if they are handled a lot or bathed just prior to a feeding. Again, they only have so much energy and need to conserve.
Positioning is important. The impulse is to snuggle the baby in the crook of your arm to feed him. This is a bad idea with preemies for 2 reasons - the baby will probably sleep if you snuggle him, and if he chokes, it will take longer to reposition him to clear his airway.
Choice of nipple can be very important. We have 4 different types of conventional nipples ( and a couple of others for babies with clefts). First is the standard nipple used for healthy term babies. I use that for most preemies as well. Then there is the "orthodontic" nipple which allegedly is more like feeding at the breast. I've never been convinced, but the nipple is quite useful for a subset of babies who choke or dribble, since it places the breast milk or formula in a different spot in the mouth. Another is softer than the others and has a larger hole. It is useful for feeding babies who have the swallow and breathe portions down, but who have a weak suck. Sometimes I use this one to finish a feeding if the baby starts to tire. The last nipple is a "slow-flow" nipple. That one is useful for the babies who have a really strong suck, but are inexperienced with the swallow-breathe portion of feeding.
Breastfeeding has a separate set of challenges in near-term preemies. Most babies take some time to develop an effective latch and suck due to a disparity between baby's mouth size and mom's nipple characteristics. It's hard for the baby to get enough of the areola into the mouth. There are ways to deal with this if the disparity isn't too great.
Mom may need a second set of hands to get the baby positioned and the mouth opened wide enough in some cases. Once the baby latches, mom needs to pull the baby in closer while she (or someone else) presses down gently on the chin to get the mouth open wider. As with term babies, it is important to make sure the lips are turned out, not in. In extreme cases, it may be necessary to use a nipple shield (silicone is best according to our lactation consultants) until the baby grows a little. I encourage the mom to try without the nipple shield first - only for a few minutes if it isn't working.
Dribbling and choking aren't a problem for breastfeeding babies unless the mom has an especially vigorous let-down. That can be managed too. We sometimes have the mother pump her breasts for a few minutes before letting the baby nurse. Estimating intake can be an issue. In most cases, we just weigh the baby's diapers to make sure he isn't getting dehydrated. In special circumstances, the baby is weighed before and after feedings (dressed and diapered). That allows an estimate of intake if that is necessary.