Well, to be fair, most of the Roman expansion occurred southwards(for example, North Africa was considered far more crucial to Roman interests as it was the main breadbasket that fed Rome).
Sure, they expanded southwards also, and I think that the general tendency of southward invasion does hold--just thought I'd add that there are some important exceptions to the general rule. There are others too, such as a biggie that I momentarily forgot--the Arab Islamic empire that invaded north into Europe as well as West into northern Africa on multiple occasions.
One of the differences is, invasions that originate in "barbaric" areas, such as the "Viking" invasions of central and southern Europe from the north, tend to sack and pillage the centers of agrarian or mineral wealth or treasuries of gold, jewels and metalwork, and then return north to their homelands shortly thereafter or after holding them for relatively brief periods with loose administration, or controlling only certain prime trading cities (such as Kiev and Dublin by the Vikings) whereas the invasions that emanate out of the agrarian centers (Rome, the Egyptian Nile, the Tigris-Euphrates, the Chinese river deltas, etc.) tend to involve longer term conquests with more intensive administration and centralization of authority.
In both cases of southward and northward invasion, the plentiful resources of the river delta areas that you mentioned do play a key part. The barbarians invaded south to get the food and agrarian-created wealth and the empires invaded north and east-west by using the food and wealth they accumulated to hire standing armies.
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On second thought, from my memory of history, the most common invasion direction was probably from east to west, rather than north to south, as one "barbarian" tribe after another left or was pushed out of Asia and Eastern Europe, often following herds of cattle and/or game, and invaded or aggressively migrated to Central and Western Europe (Celts, Scythians, Goths, Tatars, Huns, Mongols, etc.), but on the north-south axis I do think that southward invasion was more frequent, though less long-lasting and having less long-term impact on societies than northward-and-westward conquests (for example, Celto-Germanic nations like France, Spain and the British Isles now speak Romanized languages rather than vice-versa).