Inability to Cross Water
The vampire may not cross running water, except at the ebb and flow of the tide. He may be carried over or at certain times he may change shape and fly or jump over. This is not to say that they cannot swim, but running or flowing water such as rivers, streams, or waterfalls mystically impede the creature’s ability to swim and stay afloat, causing it to drown and perish. This is but a temporal "death", however. Once a vampire’s body is removed from running water, it will return to "life".
"It is said, too, that he can only pass running water at the slack or the flood of tide." - Professor Van Helsing in Mina Harker's Journal, Chapter XVIII of Dracula by Bram Stoker
"The Count, even if he takes the form of a bat, cannot cross the running water of his own volition, and so he cannot leave the ship." - Professor Van Helsing in Jonathan Harker's Journal, Chapter XXV of Dracula by Bram Stoker
It is not known how Stoker arrived at this notion. Perhaps he invented it. But there are precedents for this in folk beliefs.
On some of the Greek islands, including Hydra, Kythnos and Mitylene there was occasionally found the practice of re-burying the corpse of an alleged vampire on a desert island in belief that the vampire could not cross the water to another shore.
In his book The Customs and Lore of Modern Greece (first published in 1892, reprinted in 1968 by Argonaut, Inc.), Rennell Rodd wrote:
"Hydra is said to have been formerly infected by vampires, but a zealous bishop transferred them to the unoccupied island of Therasia, in the Santorin group, where they still walk at night, but being unable to cross salt water, find no one to torment."
One case of this practice on the island of Kythnos recorded by Henry Hautteweur in his Le Folklore de l'Isle de Kythnos (Brussels, 1898) is translated on pages 268-70 of The Vampire in Europe by Montague Summers, first published in 1928.
Here the vampire broukolakas) is a dead man named Andilaveris who terrorized his village every night but Friday, the only time when he must rest in the grave. Finally the village priest, the night watchmen of the cemetery and Church, and some other people exhumed the corpse on a Friday night, put it in a burlap sack, and transported it by barque to the tiny, unihabited island of Daskaleio. But, at some point on this journey, apparently after they reached the islet, the vampire revived from his slumber and attacked the priest by throwing mud and excrement. But somehow they managed in the end to bury him at a remote spot on the desert island.
Montague Summers gives another example in his earlier book, The Vampire: His Kith and Kin, first published in 1927. This one is taken from Travels and Discoveries in the Levant, Volume I, p. 213, by Newton (London, 1866). Summers wrote, as one complete paragraph, the following:
"Newton....says that in Mitylene the bodies of those who will not lie quiet in their gravcs are transported to a small adjacent island, a mere eyeot without inhabitants were they are re-interred. This is an effectual bar to any future molestation for the vampire cannot cross salt water. Running water he too can only pass at the slack or flood of the tide."
Cases of vampires unable to pursue someone any longer after the latter crossed running water are also found in Chinese tales.
This general belief is also applied to other monstrous creatures like fairies.
"If chased by evil fairies, one could generally escape by leaping to safety across running water, particularly a southward flowing stream." - Katherine Briggs - An Encyclopedia of Fairies (Pantheon Books, 1976), p. 336
RAW, I think it is unclear and thus it would be up to the DM. It is not absolutely proven that "flowing water" and "running water" are intended to be equivalent specifically for the purposes of vampire weaknesses.
