All central banks purchase each other currencies, especially the dollar. So when Euro drops and dollar gains, the EU central bank just bought a bunch of dollars off the open market. A little country like Poland holds $88B, now that dollar has risen probably a lot more. This is one the par with its entire GDP. Of course all the central banks just print the money, so its all inflation. That is what people mean that US #1 export is inflation, central banks prop up the dollar by inflating their currencies.
The thing is, that most central banks can't keep up pace with our feds inflation which is at 18% this year (see shadow statistics website). The dollar has been in danger for some time. A few weeks ago EU central bank raised its rates .25, but a few hours later issued a statement saying they would raise rates again. This was key. There was a real threat that there could be a run on the dollar or at minimum the dollar loosing its reserve currency status. That is what prompted their statement and retraction of the policy. Looks like the banks are taking steps to prop up the dollar, at the expense of everyone else, as it was getting too close to falling off the cliff.