As Riley said, some kids as young as 5+ may need deodorant, sometimes... their hormones are fluctuating a lot.
My daughter is 6, and sometimes needs it too.
But for her, we only use "deodorant" stick, natural kind. Not antiperspirant.
Per the 10 year old.... do NOT use aerosol.... and if he is constantly applying it... well that is too much! And if he is smelling it, and IF it is an aerosol type... he can get inhalation problems. (some kids, use aerosol cans to purposely get high, ie: "huffing."). The 10 year old... should be "taught" HOW to apply it, AND how often, AND about the side-effects from applying too much. ie: his headaches and sniffing it. Probably by accident, he is getting inhalation side effects from it.
ALSO, though, if he does NOT need deodorant yet... then he should be taught that he can't just do and use anything his older brother does. There is a great difference between 19 and 10 years old. Or, perhaps just get him an all "natural" deodorant stick, from a natural food store. And if he can't use it properly, then he isn't 'ready' to use it.
All the best,
Susan