Is it appropriate to send out a manuscript under review to a professor? I have a MPhil and planning to start my PhD soon. I sumbitted a manuscript to a popular journal in the field. After two rounds of major revisions, the editor is requesting a minor revision now. I finished the revision and submitted. The revised manuscript is now "with editor". 

At the same time, i am emailing an oversea Professor who is one of the big names in the field. I am trying to invite her to be my co-author of my second manuscript. She agreed to read. And she asked me to send her the one with editor. Is it appropriate to send her the manuscript which is "with editor"? She mentioned that one of her PhD students may want to read my manuscripts too.

In fact, i would like to collaborate with this professor and ask her to be my oversea PhD advisor. I am happy that she wants to read my works. But i am not sure if this is an appropriate action to send out a manuscript under review. 

Thanks in advance.
You still hold all rights to your work, so yes, it is fine to send it. 

I'm assuming that you don't have concerns that your work will be stolen if you show it to someone else. 

Even if you sign away your copyright to the paper, no one is ever likely to object to informal sharing with colleagues.