Naturally, your nasal passageways and sinuses drain down into the back of your throat - and you swallow the drainage without even realizing it. In the diagram below, it's the oropharynx where your mouth and your nasal/sinus passageways come together.
Sore throats are a common symptom of the common cold. You may notice the pain more when swallowing the ...
If you feel a cold "coming on" then your body (immune system) is already fighting it.
I was unable to find studies on gargling to prevent respiratory illness.
However the AAFP released guidelines and recommendations for the common cold that looks at many studies for evidence for
Decreasing the length of illness symptoms
I don't think gargling could prevent sickness. Flu and cold are caused by viruses and they usually enter from the nostril. That is one of the reason why facial mask cover the nose as well. Regular gargling with Listerine can suppress the bacterial that grow on your teeth gaps but in the long term it will alter the bacterial ecosystem in your mouth.