An answer that works is to make your own environment variables, setting those apart from language variables, and as needed for each of your multiple sites.
Then you can use these variables to make full paths for your Asset definitions.
You'll need definitions for both the file system path and URL path. Let's name them appropriately so we don't get confused with {siteUrl}, which is common in a multilingual config or {{siteUrl}} which is the Twig variable resulting from whatever sets it for a given Craft site for templates, but not usable within CP configuration.
Here's an example config:
return array(
'cpTrigger' => 'yourTrigger',
'environmentVariables' => array(
'siteStore' => '/storage/yourId/www/yoursite_html/',
'sitePath' => 'http://yoursite.com/',
),
'siteUrl' => array(
'en' => 'http://yoursite.com/en/',
'de' => 'http://yoursite.com/de/',
'fi' => 'http://yoursite.com/fi/',
),
);
In the CP on a multi-language site, we'd fill in Assets->Sources->Local Images->File System path as:
{siteStore}yourResources/images/
And to match, the URL would be set to:
{sitePath}yourResources/images/
With this, your paths for images and their generated transforms will be stable and correct across languages as well as the default URL.
To extend across multiple development, staging, and live sites, you'd just replicate this extra config in the usual way, filling in suitable siteStore and sitePath values for each environment.