I know Urgency is a special action so you have to use the points immediately, but what if the timing window is interrupted with Slippery Target or Parting Blow or something? Do you lose the movement points? Same question for Onar's Rush ability

Parting Blow can interrupt the current ability when it triggers, and then you continue resolving the ability afterwards.

Special action is not a timing window. You just resolve the special action. There are not really timing windows (other than "during your activation", which is outside of actions). There are triggers. And spending movement points during Urgency provides a lot of triggers, one at a time.

What the special action does is make sure you resolve all parts of it before you reach the end of the special action, and can then perform other "during your activation" abilities.