You have to differentiate between Rosh Hashana, when a minority of people do not wear a Kittel and Yom Kipour where almost all the married men (not in their first year of marriage and not in Aveliuth) wear a Kittel.
On YK it's rare.
So, on Rosh Hashana, you will see long black jackets, bekishehs and other similar attire. I assume you're using bekisheh as a catch-all for fracks, long jackets, bekishehs, shtrockes and heavens knows what else they wear and call them.
Whether a kittel is worn over the long jacket or instead seems to be a personal preference. (Possibly based on your attitude to A/C and how well it works in your shul.)
Never seen anybody wear the kittel under their jackets, except at a Chuppa.
But, Kittel is a vague term in this context.
While many will exchange their long jackets for the classic kittel, others - Chassidim, usually, but not always - will opt for a white long jacket or white bekisheh.