I've read some bits and pieces over the years that tampons were fashioned out of materials such as papyrus and soft wool, although sourcing that material is problematic so I'll just go over what is readily available, which would be Biblical commands regarding such matters that would apply to women of the Israelite tribe(s):

Leviticus 15:19-31

19“ ‘When a woman has her regular flow of blood, the impurity of her monthly period will last seven days, and anyone who touches her will be unclean till evening. 20“ ‘Anything she lies on during her period will be unclean, and anything she sits on will be unclean. 21Anyone who touches her bed will be unclean; they must wash their clothes and bathe with water, and they will be unclean till evening. 22Anyone who touches anything she sits on will be unclean; they must wash their clothes and bathe with water, and they will be unclean till evening. 23Whether it is the bed or anything she was sitting on, when anyone touches it, they will be unclean till evening. 24“ ‘If a man has sexual relations with her and her monthly flow touches him, he will be unclean for seven days; any bed he lies on will be unclean. 25“ ‘When a woman has a discharge of blood for many days at a time other than her monthly period or has a discharge that continues beyond her period, she will be unclean as long as she has the discharge, just as in the days of her period. 26Any bed she lies on while her discharge continues will be unclean, as is her bed during her monthly period, and anything she sits on will be unclean, as during her period. 27Anyone who touches them will be unclean; they must wash their clothes and bathe with water, and they will be unclean till evening. 28“ ‘When she is cleansed from her discharge, she must count off seven days, and after that she will be ceremonially clean. 29On the eighth day she must take two doves or two young pigeons and bring them to the priest at the entrance to the tent of meeting. 30The priest is to sacrifice one for a sin offering and the other for a burnt offering. In this way he will make atonement for her before the Lord for the uncleanness of her discharge. 31“ ‘You must keep the Israelites separate from things that make them unclean, so they will not die in their uncleanness for defiling my dwelling place,b which is among them.’ ”

So, in short, this group of ancients considered the natural cycle of menstruation to be essential vile and those experiencing it to be unclean, isolating the women during their cycle from everybody else (other than women who were also menstruating) for fear of making them unclean as well through physical contact with them. A ritual bathing was then implemented that was intended to allow them to reenter society free from their "contagious" impurity.

Along with this, is there any insight as to how cavewomen dealt with this? Did they just let it stream out or something?

If I could ask a related question, is "Menstruation = impure" concept a commonly held one in human societies, from a historical perspective?