Always found if I gargled luke warm water before a competition and also just sing a scale, it clears my throat and I'm usually ok to go. Totally agree about breathing from the diaphragm, always find if I'm running out of air in a line, just a little tug in of the stomach will help to just get that last word out without sounding like you are gasping.
For teaching children, I use little ditty songs for warm ups such as Kookaburra and Speed Bonny Boat, they enjoy them and it warms up voices and works on diction and breathing too