Hello acmoehnke -- it looks like you're asking about advanced quota options which is covered using the link below. I think you should be able to address that use case. (Word to the wise which I had to figure out in process: to enable quotas to work you need to associate them with a piece of embedded data, like the display of an element that all respondents will see. So you would need to embed data for each block for which you're interested in creating a quota.) I hope that helps!
Quotas (qualtrics.com)

Thanks Adam for the response. I've read through the online manuals, but I'm still pretty lost on this one. Tracking my understanding:
1) First, set each of the 6 blocks as its own embedded data
2) Establish a quota for each block that records if a response is submitted for that block (Embedded data--block name--is not empty)
3) Create branching logic that routes respondent to block with lowest amount of responses --> struggling here with the original question, which is how do I route to lowest amount of responses, and not an absolute number?

acmoehnke I follow you and that sounds about right -- I'm not sure exactly how Step 3 will work in terms of routing to the lowest number of responses. I'm wondering if it may require some Javascript code, which I'm not very familiar on -- but I'm upvoting this in case someone else has some more ideas here...