Tossing into the melting pot, I was taught that "Yoroshiku Onegaishimasu" is translated as "please do me a favour", or "please look kindly on me". "Onegaishimasu" is the shortened version but is used in the context "please look kindly on me". When I was in Japan, at my local dojo everyone said "onegaishimasu" at the first rei, and every time you started to work with a new partner (which was each new technique). At the Kobe dojo I visit often, they have the same formality. At the Aikikai Hombu, the same, but you stick with the same partner, my memory is fuzzy here, but I recollect saying it for each technique, but that may just have been me.
I don't recollect anywhere that seniors didn't say to juniors in normal practice, but I do recollect that if you partnered up with someone of a high level (6th Dan upwards), some of the Shihan just bowed, the onegai if said, was inaudible.
If we were changing partners, we would say "arigato gozaimasu" at the end of training with each partner.