I take it you are talking about drug lords being extradited to the states.
In their own countries a drug lord holds lots of influence. Look at Pablo Escobar for instance. he was arrested and placed under house arrest. Did you see how he lived. It was a palace and he could do anything he wanted. He had parties and nothing really changed for him.
If he was extradited to the states he would have no privileges and would spend the rest of his life in prison. He would be treated as a criminal and have no special treatment. It doesn't matter how much money or influence you have in another country as this is all foreign to you. No one cares as you are just a number with no chance of parole and you will die away from your friends and family.
Well I'm pretty sure it's along the lines of either not wanting to be killed by rival drug barons or not wanting to have to spend the rest of their foreseeable live waking up to grey bars and an orange jumpsuit.
These people have committed unspeakable crimes in their home countries or wherever they could be potentially extradited to so they know if they get sent back there it's over for them, that's why they're so scared of being extradited.
Besides, extradition means that they lose their freedom, that's probably why they leave the country in the first place, so that they can still hold on to it. If they get extradited then it's either the grave or a prison that awaits them. If it was me, I'd be terrified of being extradited too.
I assume you're talking about extradition to the US, right?
I think it's just a matter of comparing the life they would have in a local prison vs a US-based one. The US prison regime is definitely one of the toughest in the world. Money might buy you extra commissary items, but you will not have any sort of freedom while you're there.
Compare that to some South American prisons, where guards can easily be bought. You're able to live a much easier life there, as long as you have the money to pay for it, which I presume drug lords have ample access to.