There are some factors, specific to individual border regions, that incentivized and constrained the furthest adventures, as well as the more enduring borders, of Persian and Macedonian kings in common. There are even some very odd resemblances; Darius I briefly fought near the Danube on the West side of the Black Sea, in the same area Alexander led some of his first, less dramatic, campaigns. Of course, there were significant border differences as well, especially when one compares the widest Teispid/Achaemenid borders to those of Alexander’s somewhat brief Argead Empire and the more lasting Seleucid successor state boundaries to the stable borders of the Persians.
To begin with, Alexanders campaigns took him through Asia Minor by force of geography and then saw him target the major Achaemenid provinces. The old Phoenician cities and the Nile Delta attracted Alexander and the Great Kings for their naval and economic power. Mesopotamia and the Persian royal capitals were a clear target for a Macedonian who looked for total victory. The pursuit of Darius III and then Bessus led Alexander to the Bactria region and the upper satrapies, but these regions were also rich and well populated in this period. Even when he went beyond the borders of the contemporary Persian empire, such as the campaigns in the Indus valley, his decisions could be compared to those of Darius I, who had temporarily made that region an Achaemenid satrapy. Again, likely both conquerors were lured by the presence of well off settled cities and both were constrained by the Hindu Kush and, later for Alexander, the psychological boundary of River Hyphasis.
There were differences though. Darius I allegedly campaigned far past the western deserts of Egypt into Libya, while Alexander never went past siwa. The Argead empire obviously controlled regions of its Macedonian homeland and Greek backyard with levels of control the Persians never exercised. Ancient borders in general could be difficult and tenuous as local dynasts could exercise widely varying degrees of sovereignty in remote areas. So, it will likely be most valuable to look at a few regions where borders were similar over the last six centuries BCE and how those specific borders were made and maintained.
The campaigns and ultimate borders of the Achaemenids and Macedonians in roughly Central Asia were probably the most comparable. The conquering Great Kings Cyrus I and Darius I had fought difficult battles in the region of the Oxus and Iaxartes long before Alexander did. Cyrus allegedly died in the region and Darius was very proud of defeating someone he called Skunkha the Scythian, probably just a bit west of the Aral Sea. Darius’s Behistun inscription was surprisingly meaningfully modified after initial carving to incorporate this victory. Alexander was perhaps most successful beyond the Iaxartes; in that he apparently concluded an opposed crossing of the river before rapidly withdrawing to deal with a revolt in the city-dwelling areas of the region he saw as more valuable. In the Achaemenid empire crown princes would often control this region when a royal campaign wasn’t in progress. This probably indicates that the settled cities were a valuable seat of power, but also that the military situation merited the closer attention of the sons of Achaemenes.
Now the ‘stable’ borders of the Argead empire really weren’t stable, and shortly after Alexander died the northeastern front sort of exploded. Some level of control was re-exerted by Seleucus Nicator, who ultimately ruled the Seleucid state, the successor empire most comparable to that of the Persians. More specifically comparable to the Achaemenids, the son of Seleucus I and second king of the dynasty Antiochus I was made co-ruler in his father’s lifetime and ruled, inter alia, the upper satrapies. The Greco-Macedonian Seleucids attempted to impose some order on that border using similar ideas to Alexander, re-founding Alexandria Eschate as Antioch in Scythia and stressing the boundary altars along the border. The fluidity of the borders here is illustrated by the gradual secession of the local dynasts from the Seleucids, who still controlled Iran, sometime not 100 years after the death of Alexander. When another conquering Seleucid king, Antiochus III, showed up in Bactria, he agreed to a limited diplomatic solution with the Bactrians, after significant efforts to subjugate them. In this way Seleucid borders were later fixed by a political force the Achaemenids never encountered, an independent, settled, and powerful Hellenistic kingdom. It should also be noted that the border around the Caspian Sea was fixed by efforts to control points east of Parthia as well, as the Caspian gates near the Southern shore controlled access to Bactria/Sogdiana and areas around them. This proved a minimum northernmost point of control for empires desiring to control Central Asia, and when the Seleucids permanently lost the old satrapy of Parthia to the Parni it spelled the end of their eastern control.
The Indian borders are very interesting as well. As mentioned previously, bits of probably the Indus valley were the Persian province of Hidush after campaigns by Darius I, and this was probably the most successful anyone out of the Middle East was at lasting control of this region in this period. Alexander didn’t have a lot of time, or probably energy, to dedicate to setting up lasting and adaptive structure of political authority in his empire, and by the time Seleucus was consolidating his empire Chandragupta Maurya was a major new force in the region. Seleucus campaigned in India, but the records of the event speak mostly to the peace treaty he signed with Maurya. The Macedonian successors to Alexander and the Achaemenids ceded regions the Persians had controlled, possibly up to a border after the Gedrosian desert, to the Indians in exchange for 500 Indian war elephants and a surprisingly lasting peace. Thus, even the furthest border of the consolidated Macedonian empire most comparable to the pre-conquest Achaemenids was significantly withdrawn from the widest extent of Persian control and it was withdrawn because of a mutually beneficial treaty with an entirely new power in the region!
Later, the warrior king Antiochus III would march to India and find the power in control on his border much diminished from Chandragupta’s height. Instead of attempting to restore Darius’s border from his position of strength he chose to again ratify the current border and treaty with the north Indian state he found, in exchange for yet more elephants. In this way the Seleucid king most like Seleucus himself chose to play the role of Seleucus at the fixed and historical border of his realm, instead of the role of Alexander or Darius I. And so, the Seleucid elephant core, which fought so successfully at Ipsus and so…not successfully…at Magnesia was fed by Indian war elephants, themselves a product of long-term diplomatic relations at an entirely new border.
I’ve brought the Seleucids up twice now as the Macedonian equivalent of the more lasting bits of the empire Cyrus made, so it may be worth discussing the central homelands of both powers. Persian royalty could travel on a system sometimes called the Royal Road. The main artery reached from the city of Susa in Iran to Sardis in Asia Minor, passing through Mesopotamia and Babylon along the way. The Seleucid empire started in Babylon after Seleucus returned to the city he used to administer as a force in his own right. Bits of the Persian Royal Road were then used by early Seleucid kings to travel through their empire, which stably controlled the Iranian heartlands of Cyrus and Darius. The Seleucid Royal travel still included Sardis, went east to the newly important Seleucid-founded cities of Syria, followed the Euphrates to Babylon and Mesopotamia in general, and deemphasized the Iranian cities, though Ecbatana was still a frequent-ish stop.
The four cities of the Syrian tetrapolis along that road formed a sort of capital for the Seleucids, a major change in human and economic geography from the Achaemenid times, and a new border themselves. The split between Ptolemaic Egypt and the Seleucids was maybe the biggest adjustment to Persian borders. Even if the rebellious Egypt proper is not considered stably under the control of the Achaemenids, they did own Syria and the Levant, which were split between the new Hellenistic powers along a fairly contentious border. In fact, six major Syrian Contentions resulted from this split and caused significant changes in control. Again we see city founding play a role in Syria, actually a much, much larger role than in the Northeast. This both made the region more defensible, but also made it the sort of comparatively coinage-rich, settled, region that would have drawn the attention of empire-builders like Cyrus, Alexander, and Seleucus.
Finally, Seleucid presence at the western end of their major road network (now taking a different route than the Road of the Persians) proves an interesting case. The rulers of Persepolis had exerted some control over the cities of Asia Minor, though it could be somewhat changeable further from its center in Sardis. The Seleucids controlled this region at some points as well. When Antiochus III Megas went to reassert this control he justified himself with claims of historical precedent. Not that he wanted to seize old Achaemenid borders on this front or any other, but that Asia Minor belonged to the Seleucids because of the short-lived victory of Seleucus over a Macedonian rival at Corupedium in 281. Again as at the Indian border, the (literally) Great Seleucid conqueror had an idea of where his empire should extend to that led him to renounce conquest in the East and pursue it in the West. But this idea was conceived on Seleucid terms, not Achaemenid ones.