Running water in this sense are streams and rivers which served as a natural marker boundary for local regions.
There is an element in vampire law (for example the need to sleep on their home soil) of them Not being able to leave their home area. So, crossing borders would be breaking that law and that would result in death.
Why Can’t Vampires Cross Running Water
In European folklore water was considered a barrier that stopped the crossing of unholy beings, but running water was believed to be the strongest in this sense.
Since running water is usually much cleaner and less likely to harbor diseases, it was believed to be holy. Vampires are unholy, wretched and disease-ridden abominations and the purity of these waters repelled them.
The same also applies to witches in European mythology. I believe it has much to do with the religious view on the natural world.
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Why do Vampires Hang Out in Swamps?
Vampires live in mores and swamps because the water there is stagnant – and usually unclean.
Can running water Hurt Dracula?
Running water is NOT a weakness for Count Dracula.
Can Vampires Cross Running Water, Or Enter A Home Uninvited?
That would depend on the type of vampire and its age. For Dracula, he can run across running water.
However, vampires who are not directly sired, and was turned in say in the last 20-35 years would be unable to cross running water.
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Why Can’t Vampires Cross Thresholds?
As for invitation, that is up for debate. Some can cross one’s threshold without an invitation while others cannot.
Why Can’t Vampires Enter Uninvited?
Vampires might not be able to enter the front door uninvited but can use a window or back door. At Christmas they can also use the chimney if you have one.
They can’t step in the water but they can jump over it or fly.