The flash was originally designed for showjumpers, in my understanding, to offer the bit stabilization and jaw control of the drop noseband, along with a good attachment point for a standing martingale at the cavesson portion of the noseband. Ideally, not for "tying the mouth shut" unless there's some extreme, irreparable reason the horse is gapping, it's more for stabilizing "sloppier" bits like loose rings or french links. Rarely is it truly necessary to ride a horse who has had a good foundation and understanding of the aids in a flash bridle. They're currently pretty fashionable to be used as a "quick fix" for horses ridden from the front back, or that evade the bit. Watch your discipline rulebook for how legal they are in your events if you're competing... hunter classes I'm pretty sure have issues with them in competition.
As far as introducing it, I'd just start with it fairly loose and gradually tighten it to the correct tension as the horse acclimates. If it's so tight that the cavesson band is being pulled down, it's too tight and can restrict breathing. You should be able to slide your finger around the band, and fit 2 fingers between the band and the horse's face.
A stubborn horse walks behind you, an impatient one in front of you, but a noble companion walks beside you ~ Unknown