We know disappointingly little, but the usual understanding is that their power waned the further into the steppe you go. It's also important to note that most maps of pre-modern empires greatly exaggerate their control. Well defined borders are a very modern concept outside of a few cases of obvious natural boundaries like rivers. Ancient cartography wasn't overly accurate, or even very common, and in open plains like the steppe, it's basically impossible to define an arbitrary boundary. For your reference: this link is my preferred map of the Persian Empire at its height; this map is the best for showing both the theoretical boundaries between provinces and how some maps can exaggerate territory (compare the borders to the previous one); and this one shows the timeline of Persian expansion.

A large part of the problem is that information about the Achaemenids in general kind of dies out between Xerxes I and Darius III. Obviously, that's overstating it a bit, but we know exceptionally little about what was going on east of Babylon after the reign of Xerxes. Greek Historians after Herodotus either didn't include much about Persian internal history, or their works weren't prioritized and copied by later ancient historians/scholars (specifically, I'm thinking of Ctesias). After the reign of Xerxes, we also lose the most important Persian primary source: the Persepolis Fortification Archive, which doesn't actually say much about the eastern provinces, but does give hints to general demographics.

So what do we know? We know that many of the Central Asian provinces on those maps were incorporated at least by the reign of Cambyses, but most probably under Cyrus the Great. This list of provinces from the Behistun inscription lists all the territory controlled by the Persians (at least nominally) at the beginning of Darius the Great's reign, and thus at the end of Cambyses' as well:

Persia, Elam, Babylonia, Assyria, Arabia, Egypt, the countries by the Sea, Lydia, the Greeks, Media, Armenia, Cappadocia, Parthia, Drangiana, Aria, Chorasmia, Bactria , Sogdia, Gandhara, Saka, Sattagydia, Arachosia, and Maka; twenty-three lands in all.

I've bolded the ones that were part of the steppe, or had step territory. "Countries by the Sea" is a little vague, so I've marked it as well. In general, this seems to be interpreted as land either on the Mediterranean coast, or the Aegean islands, but it may also refer to the the area on the eastern Caspian Sea, marked as "Daoi" on the maps. In this list, the area of Margiana is considered part of Bactria.

Cyrus the Great, continued trying to push into the steppe literally until his dying day. According to Herodotus, he conquered the Amyrgioiand died trying to conquer the Massegetai, a Saka tribe part of wider group later inscriptions refer to as the Sakâ tigrakhaudâ. These conquests probably meant defeating them and then enforcing terms of surrender. These terms would include payment of tribute and military alliances with the Persians, and probably installing a friendly king/chieftain over the defeated tribe/confederation. This is actually explained a little in the Behistun Inscription when Darius describes his defeat of the Sakâ tigrakhaudâ (my translation uses Scythian instead of Saka, they are equivalent):

King Darius says: Afterwards with an army I went off to Scythia, after the Scythians who wear the pointed cap. These Scythians went from me. When I arrived at the river, I crossed beyond it then with all my army. Afterwards, I smote the Scythians exceedingly; [one of their leaders] I took captive; he was led bound to me, and I killed him. [Another] chief of them, by name Skunkha, they seized and led to me. Then I made another their chief, as was my desire. Then the province became mine.

The final region of Saka, the area east of the Caspian Sea, is the least well understood. The positions of the other two Saka groups on the map are based on descriptions of their geography, but it seems odd that that obvious coastline was allowed to plunge so deep into Persian territory without being conquered. This area is labled "Daoi" on the maps, based on much later accounts of where the Daoi lived, but the Daoi are not mentioned anywhere until the so-called "Daiva Inscription" of Xerxes I. So at some point they were defeated like their neighbors, but the date isn't clear.

Sometimes, the Persians would claim territory, especially distant territory that they didn't truly control. The best example of this is Darius the Great. In 513 BCE, he invaded eastern Europe and chased a retreating Scythian force deep into the Steppe, they refused to fight because they knew they could win by just overextending the supply trains and eventually Darius turned back. Yet, for the rest of his life, Darius claimed "the Saka Across the Sea" as his subjects. Xerxes dropped that charade when he came to power.

After Xerxes' failure to conquer, Persian foreign policy appears to have shifted. Future kings did not seek to conqueror new territory, but instead to influence their neighbors through trade policy and financial support/aggression, while consolidating internal power and trying to maintain the gains won by their forefathers. Thus no further expeditions into the steppe are recorded. Darius' failure in eastern Europe also probably soured any desire to expand further into the grassland. As it was, they only maintained loose influence on that frontier, through loyal but largely independent local leaders. They largely maintained the status quo for the Persians. Even then, local leaders sometimes turned against the Persians, or couldn't entirely monitor their whole territory and Saka horse-archers would raid the northeastern satrapies.

Part 1/2