To run vs to rush

Some examples from the web:

• There's really nowhere to run.
• You said that one of things driving you To run around out there was that you couldn't help your dad.
• That's when I started to run.
• Please don't try to run.
• To run the bike shop, we...
• To run an errand for the commodore.
• To run the girls' soccer program.
• To run a new set of taps... highly necessary.
• And they hired one of the top geneticists To run their program.
• You probably just wanted To run away from it all.
• To run the most elite division of Detectives in the L.A.P.D.
• My nightmare, my loop, has been To run statistical probability scenarios For where the technology might lead.
• An illegitimate child would've have ruined his plans To run for governor.
• To run off with a pair of initials.
• Specifically, I'd like to discuss How you used them to convince Skye To run off and rescue Lincoln.
• To run, to stay, live or die.
• To run away would be more accurate, 'cause I can't stay here.
• To run up four flights of stairs.
• To run a horse breeding operation, you need a certain touch.
• To run a full service state of the art soup kitchen.

Some examples from the web:

• They had to rush him here in an ambulance.
• I hate it when they have to rush these things.
• No reason to rush this relationship.
• Tell the lab to rush the results.
• I'm not asking you to rush or replace.
• Then they told us to rush to the hot springs 'cause that's a sure-fire cure.
• I might be able to rush it along.
• You didn't have to rush over here.
• I had to rush across town to fix my dress.
• N-not to rush you, but...
• Had to rush out of there pretty quick.
• That's why I wanted to rush this wedding.
• I'll tell them to rush the tests on belladonna.
• They were trying to rush this thing.
• I decided not to rush it either.
• I decided not to rush into anything.
• Probably best not to rush her.
• You don't have to rush into marriage, either.
• I'm trying not to rush to judgment.
• I do think it's important not to rush into things.