For the Romans, at least, old coins were usually not purposefully taken out of circulation solely for political reasons. Old coins continued to circulate until they were too worn to legitimately be worth their face value (at least for silver and gold coins, whose value depending on their precious metal weight). In general, there was little reason to recall old coins for political reasons. Most new Roman rulers wanted to emphasize their connections to the past. Many of them claimed to be descendants of prior rulers despite no evidence of a connection (the Severans claimed descent from Marcus Aurelius and even took his name as part of their Reginald names). Many of them even deified their predecessors, in part to assert their own divinity.

Roman coin hoards often show a mix of rulers, indicating that coins of former rulers circulated along with the current one. The Malmesbury Hoard contained 1266 coins dated from AD 286 to 317 and which had been minted under 6 different emperors. The Hoxne Hoard contained coins of 8 different rulers over as much as 60 years from the late 4th and early 5th century. These findings are repeated with other hoards.

There were some exceptions to this. On occasion, prior rulers were subject to what was called damnatio memoriae (condemnation of memory). Damnatio involved the destruction of statues and portraits and the recall and melting of coins. This usually happened when a ruler was particularly unloved or tyrannical and the subsequent ruler wished to emphasize a lack of connection to him. The emperor Domitian underwent damnatio after his death when Nerva wished to show how different he was. You can find statues of Domitian that are intact except for a destroyed face. Caracalla had his brother Geta killed after they jointly took the throne on the death of their father and had Geta’s portraits destroyed. One of the most famous and rarest Roman coins, the Eid Mar denarius, was minted by Marcus Brutus to glorify the assassination of Gaius Julius Caesar. After Caesar’s heir Octavian defeated Brutus and the other assassins, it became considered unwise to own one of these coins, and most were melted down in antiquity.

However, even with a damnatio in effect, coins remained in circulation. Roman coins were struck in huge numbers, and recalling all of them from a far-flung empire was difficult. Neither Domitian nor Geta is difficult to acquire now despite the damnatio. Basically, it was a lot easier to deface or remove a single statue on a triumphal arch than to pull millions of coins from people’s possession.

A greater factor in taking coins out of circulation is what numismatists and economists refer to as Gresham’s Law. When faced with economic hardship, ancient governments often responded by debasing the coinage. The coins would have the same nominal value, but contain less silver or gold. This allowed the government mint to strike more coins with less precious metal. The population was not fooled by this, and citizens would preferentially hoard the older, higher silver/gold content coins and spend the newer, lower valued ones.

So the evidence would indicate that a Roman citizen was likely to have the portraits of multiple rulers in his or her purse at any given time. Nobody would have thought this odd.

See

The Art of Forgetting, Disgrace and Oblivion on Roman Political Culture (Harriet Flower)

Mutilation and Transformation, Damnatio Memoriae and Roman Imperial Portraiture (Erik Varner)