“a political process that is engaged in quickly and properly by everyone” could be rewritten:
a political process that everyone engages in quickly and properly
Catherine Ashton, the EU's foreign policy chief, doesn't like either side of the situation. On one side, you have Yanukovych backing out of one deal, signing another, then saying he's too ill to do anything else—all with impunity. On the other side, you have protesters who feel they have no other option but to engage in public demonstrations, which are dangerous in the extreme cold and sometimes turn violent.
Ashton suggests (without specifying what exactly should be changed or what it should be changed to) that some kind of new system should be put into place,1 presumably one that will give citizens2 an appropriate3 and effective4 way to voice5 disapproval with the President's actions when necessary.
1 “. . . Ukraine need[s] ‘a political process. . .’”
2 “. . . people . . .”
3 “. . . properly.”
4 “. . . quickly . . .”
5 “. . . engaged. . .”