There is generally no shedding associated with finasteride use. Shedding with minoxidil is rare. If you are concerned it is best to discuss this with your doctor.
Although there is not a good scientific reason for shedding hair after starting finasteride, it might has to do with the increase in the turn over of the follicular generative cells that are responsible for creating new hair follicles.
Rogaine rarely causes the hair to shed, if anything the hair improves long term. I suggest you see a physician experienced in hair loss treatments. Best, Dr. Emer.
In my experience, very few of my patients undergo a shed when they start finasteride. Finasteride works slowly because it works from the hair root 7 mm below the skin surface. These haira have to grow out to get benefit.