Tokens to change a block's visibility I am trying to make a block visible only to pages whose URL matches user/[current-user:uid]/downloads, but I don't get anywhere.

Do you have any idea on how to achieve this?
Fairly sure you can't do this from within the Blocks interface, but would be delighted learn otherwise.

A strategy to try would be to create a Views block that will display or not as required. A View over Users with a Contextual Filter that takes the Raw Value from URL, picking out a given path component, would be a start. You can then filter the view by User: Active.

If that all works out then your View will show. You can then get the content you want for your block by using a Rewrite of User: name field and, perhaps also, using Global: Text area in the header. (Well, you might be able to get the content this way. Depends on what's needed in the block.)

When you add the View block, you'll do something such as user/*/downloads, but the block will show depending on current user matching the contextual filter.