If you and I were discussing this over coffee I would ask you, is it the task that you object to, or the tone? Take the example of emailing something to somebody. I can think of any number of reasons why I ask someone else to email something instead of me:
- I don't want those people to have my email address and then randomize me for the rest of time when they want to ask something or report something
- The other person is best placed to answer questions that occur to the person getting the email
- I'm about to be away and won't even see replies to that email
- The other person is the official contact and it might confuse people to get an important document from someone else
Now, these reasons may not apply in your case, but the task vs tone issue is, had the director said to you:
Since you're the chief point of contact for that team, and they expect communications from you, I think it would be best if you sent this document along instead of me. I also know you'll handle any responses from them better than I could.
Would you still be annoyed? Are you irritated that this person is not doing certain things, or that this person is treating you like a personal assistant and just handing you tasks?
There are two ways things can change in the future - the director can stop assigning you tasks at all, or can assign them more politely, including with more information. It's possible that the latter might address issues with the tasks as well as with the tone, if it hadn't occurred to you that there were non-selfish non-lazy reasons for this behaviour.
So, how to get these tasks assigned more politely? Do nothing in front of clients or people who report to either of you. But when alone, or with your mutual boss, consider replying (verbally) to one of these with:
Can you explain why that's something I should be doing rather than you?
Isn't that something you should be doing?
Pardon me? (Or Excuse me, or however you clarify when you might not have heard correctly) Are you tossing me your tasks without so much as "please?"
(needs to be done with humour, could be the start of a fight, but might light a lightbulb in the director's head - sub ma'am for a female director
Yes sir, I'll get right on that sir, when do you need it by? And would you like a coffee with it?
If you apply these consistently I would hope that either the delegation would lessen or would come with more information, assuming you are in fact peers. It's also possible it will come with praise that the other meeting participants hear each time, which is also good. If you're not peers, you'll be told so in a hurry (and you may not enjoy the telling, that is the risk here.)
If these things only happen in email or in front of other people, then I would suggest waiting until immediately after the meeting or walking down the hall to the office and saying
I didn't want to say this at the time because X was there, but sending that email, isn't that something you should be doing?
(or whatever rejoinder you would have chosen.)