In my experience, Tumblr is a slow burn. It is itself a blog and the same rules apply. That said, the culture around Tumblr is a bit different.
I strongly suggest creating content that is just for Tumblr and mix it in with reblogs that your readers will love. Overall keep self-promotional content minimal. Share links to your blog at most, I say, 5% of the time. It’s hard to say as each case will be different. When you do share something it should add value to the followers you have without diluting the on-site brand you are building.