In theory at least, there is no issue in a river being considered a proper place for טבילה. Although the water is flowing, since the source of a river is from a ‘well’ or underground water, the fact that it is flowing is not a problem.
That is because ‘well water can be used even if flowing’.
The waters of a spring purify even if they are flowing. They don’t need to confined to one space ( זוחלין נמשכים והולכים ואינן מכונסים) – the meaning of זוחלין “flowing” is, continually flowing and moving and are not gathered together.
2- Rainwater that is flowing cannot be used.
מי גשמים אין מטהרין אלא באשבורן
Rain water does not purify unless it is in an “ashboren”. It needs to be limited to one space (פירוש מקום עמוק שמתכנסים בו המים ונקרא אשבורן) – (that is, a deep place in which water can collect: called ashboren).
3- Thank you to Yankee Teitelbaum who sent this link showing a flash river from rainwater.
Water flowing from a flash flood due to a strong rainfall would not be כשר to use.
4- So what is the issue with using a river? During the rain season or in the spring when the snow begins to melt the river size increases due to the rain water.
So no we have a mixture; rain water with well water. Thus, when one uses the river is he טובל in the original river/well water (no issue despite the water flowing, as above) or in the newly added rain water (which is not כשר unless it’s stationary in a מקוה).
The problem is if the rain water volume is greater than the original volume of the river.
5- In short- the מחבר – Shulchan Aruch prohibits using a river during the rain season. The Rama says that the Minhag is in many places to use it if there is no other proper mikvah.
אבל יש מתירים לטבול בנהרות כל השנה, אף בשעת הגשמים והפשרת שלגים ורבו הנוטפים על הזוחלין, משום דעיקר גידול הנהר הוא ממקום מקורו (טור בשם ר”ת וב”י בשם רש”י וסה”ת וסמ”ג); –
However, there are those that permit immersing in rivers all year long, even during the rainy season and the times when the snow is melting – and thus the majority is notfim rainwater rather than the flowing spring water, because the main part of the growth of a river is from its original source flowing from a spring. Even though it would ostensibly seem like the rainwater is causing the river to overflow or rise, it is nonetheless considered part of the river water.
וכן נהגו ברוב המקומות במקום שאין מקוה, – and so is the custom in the majority of places that do not have a mikvah to immerse in the river year round, even during the winter months, ואין למחות ביד הנוהגין להקל כי יש להם על מי שיסמוכו – and one should not object to those that have the custom to be lenient because they have whom to rely on. (מהרי”ק ובת”ה שם ומהרי”ו סימן ע’ וב”ז סימן קנ”ד).
5- We read the Sha”ch that quotes the Gemara that the increase in volume of a river during the rain season is primarily from the swelling groundwater. So basically, most of this new water volume is from well sources.