The suggestion by "Registered User" is pretty close to what I think will work. However, if you use the default GUI drill-through option in SSRS, it may not always completely reload the report, which might account for what you're finding.
If you use the "Go to URL" option in the Action tab (instead of the "Go to Report" option), you cause the browser to reload everything. My anecdotal explanation is that when you use the "Go to Report" option, you're moving around within the ReportViewer framework and so it may be using some cache to regenerate parts of your report. When you use the "Go to URL" option, I believe it forces your browser to leave the current page and re-enter the destination page.
This link has good information on report server paths and how to manipulate the URL to control how your page feels.
Here is a resource on all the different URL parameters you can use in the "Go to URL" option to control how your page is loaded and what parameters are passed through.
To use the Go to URL option, you can use the global ReportServer & ReportFolder fields so you're not typing in the full address each time. Also, if you want to make sure your parameter bar always shows up (which is what I'm guessing you're referring to by "selection boxes"), you can use the rs:Parameters=true URL parameter. FYI, one of the other great things about the "Go to URL" option is that you can integrate all SSRS functions and some limited JavaScript to really control where and how your page goes.
Hope this helps!