The previous answer does not refer to clinical studies so I thought I would have a look.
TLDR; the clinical studies I've seen suggest not to rinse with water after brushing.
According to Doméjean, et al. (2018), you should not rinse after brushing.
For maximizing the topical effect of the fluoride toothpaste, patients should be encouraged to spit out ...
Short answer: No. Using water, mouth rince, floss, toothpics or other means of removing debris between teeth should not cause dental movement.
Long answer: To cause dental movement, a continuous pression or pull has to be applied for many hours per day, over the span of weeks/months/years to make teeth move. That is why head gears have to be worn at least ...
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.
Not rinsing after tooth brushing may somewhat increase the effectiveness of fluoridated toothpaste, but the evidence is inconsistent and the effect can vary greatly among individuals.
One possible explanation for inconsistent effect: not rinsing after brushing appears to be only beneficial if you are at a high risk of getting cavities.