Many AT guns had AA gun ancestry (and not just heavy guns - this extended down to AA/AT machine guns). Not all of these were dual-purpose guns - specialised anti-tank guns benefited from low profiles, while the high elevations required for AA guns meant that large guns needed high profiles. Thus, the specialised AT version of the German 88mm gun was a better AT gun than the AA version used in the AT role - it was less visible, less vulnerable.
What AT and AA guns had in common was high velocity. AA guns needed this for altitude, and AT guns needed this for armour-piercing energy. AT shot to take advantage of this was solid shot - not so effective against aircraft, so different ammunition was required for the two roles. If there was very little chance of the AA gun needing to shoot at tanks, there was less chance of AT ammunition being available for the AA guns to be used in the AT role. Thus, many of the Allied AA-derived AT and tank guns were specialised. For example, the US 90mm M1 AA gun was the ancestor of the M2 dual-purpose AA/AT gun and the M3 tank gun (used in the M36 tank destroyer and the M26 Pershing and M46 Patton tanks. The M2 AA/AT gun was in many ways an equivalent of the German 88.
The push towards heavier tanks with thicker armour drove an arms race towards larger-calibre AT guns. At the start of WW2, AT rifles and 20mm guns were often considered adequate, but later AT guns tended to be much high caliber. High calibre and high velocity (which required long barrels) mean large heavy guns, which could be difficult to move on the battlefield. Heavy AA guns didn't need to be moved as often, but AT guns often needed to be moved on the battlefield, something that could ideally be done by the crew alone, which ceased to be feasible for the heavier guns (e.g., the British 17-pdr). With the development of shaped-charge AT warheads, velocity stopped being as important, with many effective low-velocity (and therefore short-ranged) weapons appearing (e.g., the Panzerfaust). These were portable and cheap compared to heavy AT guns. This led to towed AT guns being less important, and dual-purpose AA/AT guns also became less important.
The USSR began to develop tank and divisional guns with the ballistics of their new 85 mm AA gun back in 1940. However, none of these developments made it to an acceptable state before the German invasion.
The Red Army began to use its 85 mm AA guns as ersatz AT guns in 1941 when a shortage of AT guns became apparent. Orders to purposefully organize AT regiments equipped with AA guns were issued in the fall. This state of affairs was an emergency, as the AA gun was quite large and difficult to move on the battlefield, but these guns continued to be used this way in 1943, when it was discovered that the 85 mm gun was very effective against new Tiger tanks. This drove the development of the D-5 and S-53 tank guns with the ballistics of the 85 mm AA gun. Development of an 85 mm divisional gun was also explored, but such a gun was not adopted into service until after the war.
Sources:
http://tankarchives.blogspot.com/2016/02/experimental-tank-guns-1940.html
http://tankarchives.blogspot.com/2015/11/aa-into-at.html
http://tankarchives.blogspot.com/2015/03/overpowered-at.html
http://tankarchives.blogspot.com/2016/12/85-mm-at-gun-battery.html
http://tankarchives.blogspot.com/2014/10/85-mm-gun-range.html
http://tankarchives.blogspot.com/2013/03/soviet-85-mm-guns-vs-tigers.html