I don't think lithium or benzos can cause depression. (And benzos are pretty tolerable compared to other psychotropic meds.) Maybe he is on a downswing if he is bipolar. I thought lithium is used mainly for the manic side of BP. Lamictal (mood stabilizer) is used for hypomania (lower mania with depression) and those who tend to lean more towards depressive side of BP. If you search the web for lithium you can find answers to your questions.

Benzos can make you sleepy and/or tired and that may trigger lethargy and possibly slip into depression if they spend too much time in bed or are inactive, etc. Benzos have to be adjusted carefully for a correct dose just like any other med and can take quite a while to find that dose. Xanax is shortest acting benzo (4 hrs) and klonopin is longest acting (days) but they all do the same thing. Ativan and others are in-between. (Or more correctly, xanax has shortest half-life and klonopin longest half-life.)

I do know that when PDOCs treat BP the proper treatment is to the stabilize the manic side first then the other symptoms after that. (That's what I've read in books on BP disorder.)