to abandon a city just right click it. there you will see an option "abandon city".
when you abandon it it will turn into rubble.
it is NOT like starving a captured city. it will destory it, opening a spot for your settler to plop down. starving cities is used to lower the number of foriegn citizens in your newly aquired cities. the higher the amount of forigners in your towns = the bigger flip \ unhappiness chances.
i abandon cities if:
A: it was an AI flip\capture that i find not likely to hold onto.
B: its my city, but i am under threat and dont want the AI to get hold of an intact city. (aka as "scorched earth")
C: if i find that its a small inefficient town that would be better placed elsewhere, and which i was forced to plant in a not ideal place (like trying to grab a lux, with an AI settler 1 turn away, i would plop on spot to grab it, and later, if needed, can abandon and resettle in my preferred spot.
some cities are captured if they are over size one, have some culture or both.
if a city is size 1 with no culture it is usually AUTO RAZED.
the bigger, older the city, the less chance there is for an autoraze.
hope this helps, Jack