I want to describe a person or group of thugs who cause damage without concern for science or culture. I had written down "mongol behavior", in reference to the Mongol invasion. Is there an all-culture-friendly word to express a similar concept?
Well, as Ms A pointed out, "damag[ing] without concern for science or culture" isn't a sense that (M|m)ongol even has. The Yuan Dynasty was a highpoint of Chinese cosmopolitanism and, even in the west, the Mongols subsidized astronomical advances. Good soldiers tend to love good applied science; scientifically hostile groups tend to be religious zealots like Savonarola. Mongol's only negative connotations come from 19th-century pseudoscience, which mixed up East Asians and people with Downs syndrome.
The problem you're going to run into is that any English word referring to groups of people as unenlightened savages is going to have at least 200 years of problematic racist usage. Even the best English word for people overtly hostile to culture, philistine, harkens back to ancient racism we've traditionally considered divinely sanctioned.
Your best woke option is going to be to simply focus on the behavior itself—violent, wanton, reckless, wild, &c.—and not the people as a group. If you don't want to say some cultures are inherently superior, group pejoratives are inherently problematic.
Forget about the nationality. You say that you are thinking of the "Mongol invasion."
So, instead of using Mongol behaviour, refer to the type of activity and use invasive behaviour:
1 : tending to spread especially in a quick or aggressive manner
3 : of, relating to, or characterized by military aggression
In particular, some of Haddock's curses which might work:
If you want to avoid any cultural bias, mindless morons (she checks etymology of "moron"...) Its safe!
1910, medical Latin, "one of the highest class of feeble-minded persons," from Greek (Attic) mōron, neuter of mōros "foolish, dull, sluggish, stupid," a word of uncertain origin. The former connection with Sanskrit murah "idiotic" (see moratorium) is in doubt. Latin morus "foolish" is a loan-word from Greek.
As for the epithets mongol, and mongy, both are derived from mongoloid which used to be the medical term for those afflicted with Down's Syndrome, mongol later became highly derogatory and it is currently used as an insult to anyone whose behaviour or intelligence is considered subnormal or below average. I would advise not to use it, particularly in the context described by the OP, for that reason alone.
In reference to the genetic defect causing mental retardation (mongolism), by 1899, from the typical facial appearance of those who have it. See Down's Syndrome. Such people were called Mongolian from 1866.