Can I copy the folders from mail to my iPad?

I've been using Mail on my MBP for years and now I'm using my iPad more often and I would like to be able to see all the folders I've created in "On My Mac" in my iPad, or iPhone. I normally try to move my email to another "labeled" folder soon, so I can quickly find it and I don't drown in thousands of emails in the Inbox.

Last week I traveled and only took my iPad. I needed to read an email that I already sorted, but could not find any of the emails I had received earlier. Then I had to open Gmail and go through their system. Eventually I found them after a lot of searching.

Is there any way to these copy folders to iPad? Or should I be looking for an alternative email client?

Thanks.

 

I have a couple of Gmail accounts and one me.com (apple) account. What I have done on my MBP is created a lot of folders according to projects. I may have 50-75 folders in my MBP (under the mailbox section in the "on my mac" section) where I can easily find emails for projects I did 3-4 years ago. They are all sorted by either a business or personal, project, or family, or company etc. I like to keep my inbox to under 100 emails when possible.

When the emails are moved from the Inbox in Mail to one of these folders in my MBP, they also dissappear from my iPad Inbox. Obviously they always stay in the parent Gmail account, so if I need to find an email I have to go into Gmail to search for it.

Hope this clarifies it. Thanks.