You're asking 2 separate questions here:

1. Where did fleeing troops go?

2. Were those who fled punished?

We know a fair bit about this in Classical Greece, so I'm going to answer both. If you're looking to join this ride, bear in mind that the rout is a dangerous place, so I hope you've brought a horse or a Sokrates:

Here [at the disastrous battle of Delion in 424 BC] I had an even finer view of Sokrates than at Potidaia. (...) There he stepped along, as he usually does in the streets of Athens, "strutting like a proud marsh-goose, with ever a side-long glance," turning a calm sidelong look on friend and foe alike, and convincing anyone, even from afar, that whoever cares to touch this person will find he can put up a stout enough defence. The result was that both he and his companion got away unscathed: for, as a rule, those who act this way in war will not be touched.

-- Plato, Symposion 221a-c

 

Where did they go?

The answer here is obvious: they fled somewhere safe.

Some made for Delion and the sea, some for Oropos, others for Mount Parnes, or wherever they had hopes of safety.

-- Thucydides 4.96.7

Armies liked to fight close to their own city walls, which would provide refuge in case of defeat:

The Argives thought that they could not have a fairer field, having intercepted the Lakedaimonians in their own country and close to the city.

-- Thucydides 5.59.4

It was not even necessary for a broken force to flee within the walls, since the people left behind in a city could ward off enemies by shooting missiles and throwing stones from their high position on the city's fortifications. In this way, an army huddled at the foot of a city wall was effectively safe:

In many other instances those who have pressed a pursuit too close to a city's wall have come off badly in their retreat, and in this case also [at Olynthos in 381 BC], when the men were showered with missiles from the towers, they were forced to retire in disorder and to guard themselves against the missiles.

-- Xenophon, Hellenika 5.3.5

If the army was not near its own city, but in friendly territory, allied fortified positions were fine too:

Meanwhile the Ambrakiots and the troops upon the right wing defeated the contingent opposed to them - they being the best fighters in those parts - and pursued it to Amphilochian Argos. Returning from the pursuit, they found their main body defeated; and hard pressed by the Akarnanians, with difficulty made good their passage to Olpai, suffering heavy losses on the way.

-- Thucydides 3.108.2-3

If the army was campaigning further afield, the obvious safe ground to aim for was the camp from which it had marched out to fight that day. The Greeks usually built their camps on defensible positions, and often made rudimentary fortifications to protect them.

Yet despite the fact that many had fallen and that the Spartans were defeated, after they had crossed the trench which happened to be in front of their camp they grounded their arms at the spot from which they had set forth. The camp, to be sure, was not on ground which was altogether level, but rather on the slope of a hill.

-- Xenophon, Hellenika 6.4.14

If there was not even a camp to retreat to, the fleeing men, in their desperation, could run to the proverbial hills:

Thus the Athenian army was all now in flight; and such as escaped being killed in the battle by the Chalkidian horse and the peltasts, dispersed among the hills, and with difficulty made their way to Eion.

-- Thucydides 5.10.10

As for the Argives, they did not await the attack of the forces of Agesilaos, but fled to Mount Helikon.

-- Xenophon, Hellenika 4.3.17

While there are only a few examples of troops rallying and rejoining the fight, it is understood that those who had fled to safe ground regained some measure of cohesion. The safety of their position and the presence of their comrades would have put an end to their panic and reminded them of their duties. It would also have made it obvious that their only hope of preventing further disaster was to stick together with whoever remained. In the surviving accounts, those who reached the safety of a city wall or camp would usually request a truce to collect their dead; this ritual implies that some semblance of an army had reformed, which could speak with one voice, rather than as a bunch of little groups of survivors who had abandoned all pretense of collective action. Indeed, there are a few examples of Greeks (invariably Spartan or Spartan-trained) retreating from battle in good order. No doubt the ideal was for an army never to lose its sense that it was an army, even in defeat. The scattering of the routed Athenians at Amphipolis and Delion demonstrates the severity of their defeat; they were so badly beaten and so viciously pursued that they struggled to reform as an army.

continued below