Would it do permanent damage if I drove about 10 miles to a mechanic?

What is your question?

I had a co-workers friend install a new motor in my car. He contacted me to let me know that he accidentally added the wrong fluid in my transmission. I would just like to know if it would do permanent damage to my engine if I drove about 10 miles to a mechanic to get the fluid flushed and have the correct fluid added.