Steppe peoples were nomadic, cohesive, tribal, and wide-ranging. It is not beyond reason that they could have made appearances in all these places.
And when you say "at the same time" you don't really mean "at the same time", you mean "around the same time, say plus or minus 20 to 50 years, don't you? I'm not saying that the SAME groups of Huns atacked the outposts of the Roman Empire and outposts of China or India at different times, but as the main groups moved around it is highly likely that their presence would be felt in widely displaced locations.