I'm watching the GOP debate, and I'm noticing that some candidates are picking on Hillary.
Is there a verb for an act where a person castigates someone else in hopes of making that person his or her personal rival or enemy?
It sounds like baiting. American Heritage Dictionary defines bait as
To entice or provoke, especially by trickery or strategy: He baited me into selling him my bike by saying how much I deserved a better one.
to make (someone) unfriendly : to cause (someone) to stop being friendly, helpful, etc., towards you
Or similarly, estrange:
to cause someone to be no longer friendly or close to another person or group
These words typically indicate that at least a cordial relationship existed previously, but actions are taken to cause that relationship to cease.
Both "challenge" or "call out" fit your case extremely well; Both can be used to refer to a call to a fight, figuratively or literally. "Singled out" or "gunning for" have some of the same connotations, too.
If you're on the hunt for a term with negative denotations or connotations, you could lean more towards "bully", "sniped", or "picked a fight with".
Something along the lines of "declaim" would also fit, but has no rivalrous aspect to it.
You could say, some candidates were riding roughshod over (or dumping on) Hillary.
ride (or run or roll) roughshod over someone: to treat someone roughly or without care, respect, moderation, or control; to act in a bullying manner toward someone Wiktionary
dump on someone: criticize someone severely and unfairly The Amercan Heritage Dictionary of Phrasal Verbs
Sounds like provoking, with meaning defined by M-W under "to provoke":
: to cause (a person or animal) to become angry, violent, etc.
"Antagonize" is broadly used in that context which means:
Cause (someone) to become hostile:'It is spending astronomical amounts of money, alienating allies and further antagonizing opponents.'
Or you could consider using "make an enemy of".
Cause (someone) to start feeling hostile to one: 'you really don’t want to make an enemy of your girlfriend’s best mate'
[Oxford Online Dictionary]