After a tooth extraction, swallowing precautions are NOT required to prevent a dry socket because it's not a potential risk factor. Therefore, you may swallow normally without having to worry about developing the painful dental condition.
Rather than worrying about how to swallow there are other factors which are much more likely to cause it. I recommended focusing on those other things instead.
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Swallowing forcefully won't cause dry socket
It is impossible to cause a dry socket from swallowing because the condition is biologically induced and not mechanically caused.
The condition is defined as a socket without a blood clot. There is a misconception that the suction from forcefully swallowing can dislodge the clot thus resulting in the condition. However that is false because mechanical dislodgment of the clot cannot cause it.
Biologically induced condition
Studies have shown that dry socket is initiated via a biological process, meaning it is not mechanically induced. That means physically disrupting the blood clot by increasing intra-oral pressure such as from swallowing will not cause it.
The study which we listed above actually shows how drinking through a straw does not cause dry socket. The suction pressure from using a straw is often mistakenly believed to be able to cause it but the study has proved it to be false.
For comparison, swallow generates significantly less pressure in the mouth than drinking with a straw. The conclusion that the both of us should've reached is that swallowing should be a non-issue.
Other activities that generate intra-oral pressure:
Drinking with a straw
The exact etiology for dry socket (alveolar osteitis) is currently unknown and there are only theories as to how you get it. Nonetheless, researchers have been able to identify certain risk factors that increase your chance of getting it.
Associated risk factors:
Smoking. Studies have shown that smokers (12%) are 3 times more likely to get a dry socket than non-smokers (4%). So try not to smoke please.
Traumatic extraction. Difficult surgical extractions seem to increase the incidence of it, especially while extracting impacted lower third molars.
Birth control. Studies have shown that the incidence of alveolar osteitis was significantly higher for those taking birth control.
Middle of Menstrual cycle. In addition to taking birth control, researchers also found that women during the middle of a menstrual cycle were more likely to get it.
Previous history. If you've had this condition in the past, you're more likely to get it again. You should be extra cautious if that is the case.
In summary, if you're trying to avoid a dry socket, you should try your best to minimize the above risk factors.
It's okay to swallow
When we say it is okay to swallow normally, that includes your saliva AND blood. Some people may think that it is gross to swallow your own blood but there is actually nothing wrong with doing so.
However if the idea really grosses you out and you absolutely refuse to do it, you still shouldn't spit it out. Excessive spitting may potentially dislodge the blood clot and you'll end up with persistent bleeding. If you do need to expel the blood, you should let it drip out of your mouth slowly by gravity only. Do not forcefully spit!
Aside from your saliva and blood, there is just one thing that you shouldn't swallow and that is gauze. The gauze which you bite on is made of cotton fibers which means that it will not get digested and that will turn into a complication.
Swallowing won't dislodge the blood clot
Normal swallowing of saliva and food is unlikely to dislodge the blood clot because it doesn't create enough pressure to do so.
However swallowing by drinking through a straw can certainly create enough suction pressure to dislodge the clot. That is why it is forbidden to use a straw for the first 24 hours after your extraction. That is a part of the aftercare protocol.
From our personal clinical experience, we've never had a patient dislodge a blood clot from swallowing. We've also never heard of any cases of it happening either. Most of the incidences were due to using a straw, spitting, or rinsing within the first 24 hours.
Consequence of dislodged clot
If you manage to dislodge your blood clot, you won't get a dry socket but you will get persistent bleeding. The entire purpose of leaving the clot undisturbed is so that you don't bleed excessively. A mature clot is what makes the bleeding stop.
Therefore if you disrupt it before it matures, the socket will continue to bleed. Your top priority after the tooth extraction is to stop the bleeding because if it goes on for too long, it can be life threatening.
If the socket resumes bleeding due to a missing clot, you must consequently resume biting on gauze. That is the best way to get it to stop oozing.
Swallowing your saliva or food will not cause a dry socket because the condition is not mechanically induced. It is a biological process that occurs which is mostly out of your control.
Therefore you should rest assured that it is safe to eat and swallow your food. Proper nutrition is key to recovery and is part of the self-care required for healing. If it wasn't safe to do so, you'd need to go on a diet for about a week which doesn't make a lot of sense does it?