I just realized, could "run" in this instance refer to a political campaign as well as actual running? What made me think of this right now was "she" in this sentence being Hillary and "the man" being either Donald Trump or Bernie Sanders, depending on if you're talking about the primaries or the actual election.
Note that this difference is reflected in English too, and probably came into Irish from English - long before overt public campaigning was a factor in elections, candidates for election to parliament "stood for election".
In modern parlance, candidates in Ireland and the UK now "run" and "stand" for election at the same time - how much of this is an influence from coverage of US elections, and how much is just a reflection of the "horse-race" nature of election coverage is anyone's guess, but the NEID suggestions for this type of horse-race related election idioms ("in the running", "making the running") don't use rith:
"she's still in the running for the post" - tá sí san iomaíocht don phost fós, or tá seans aici ar an bpost go fóill
"we're making all the running in this campaign" - is muidne atá ag déanamh na hoibre ar fad san fheachtas seo
When it comes to "running a campaign", rith only turn up in one of five examples:
"how campaigns are run" - an bealach a ritear feachtais
"they ran a high-impact campaign" - bhí an-tionchar ag a bhfeachtas ar an bpobal or chuaigh a bhfeachtas i gcion go mór ar dhaoine
"the company ran a mixed media campaign" - chur an comhlacht feachtas meán measctha i gcrích
"he ran a successful campaign" - bhí feachtas rathúil ar bun aige
The OED notes that the “compete as a political candidate” meaning of “run” is “originally and chiefly U.S.” — its first appearance is in a Virginian newspaper from 1826 — though that “and chiefly” was stated before the introduction of the Web. It was listed as an evolution of the “compete in a race” meaning, such as in describing horses.
The “running a campaign” uses seem to be closer to the “operate” meaning of “run”.
?? I'm not sure what "elope" has to do with running.
The Irish for "elope" is éalaigh, which is generally translated as "escape".
For "run away", you can use éalaigh or imigh. here are some examples from in the NEID:
"she ran away with him" - d'éalaigh sí leis, d'imigh sí leis
"he ran away from home" - d'éalaigh sé ón mbaile
"she'd run away and leave it to others" - d'imeodh sí léi agus d'fhágfadh sí ag daoine eile é
Note the preposition in that last example - d'imeodh sí léi - "she'd run away with her(self)"
And just as "jogging" is not the same as "running", rith doesn't means "jog" - the irish for "jog" is bogshodar (literally "soft-trot") though you're probably at least as likely to hear ag jogáil.