Cross browser compatibility issues in WordPress are typically CSS related. You will often notice this issue when browsing your site and certain elements aren't in the right place, do not appear to resolve properly, or simply don't load at all.
Older WordPress websites could run into issues with cross browser compatibility. Due to strict plugin and theme requirements you are less likely to find any current plugins or themes with this issue.
The Microsoft Edge browser will easily help you run cross browser compatibility checks on your site.
Note: A great place to start is IE 8, since is this an outdated browser that is still commonly used.
Since WordPress itself is already optimized for cross browser compatibility, your list of inconsistencies should point to a theme or plugin that is causing the issue.