Redirect page to existing field value I am redirecting most of the pages of my old website to a new subdomain of the same domain, from http://www.domain.com/www-page_url to http://new.domain.com/new-page_url/with-different-slug. This new URL value exist in (almost) every page of the old website.

How would I proceed to create a general rule that permits the redirection from the page itself to new.domain.com/page_url existing field value?
You should be able to get this to work using the Rules module, using the "Drupal is initializing" event. That will ensure that the rule gets triggered BEFORE the actual content is shown. I'm not sure (actually I really doubt ...) if using Rules event " Content is viewed " (not " Node is viewed " ...) will work (typically that's 'too late'). Refer to "How to specify a Rules event like &quot;Content is &#39;going to be&#39; viewed&quot;?" for more details on that. 

If you can think of something to use a (new) URL like http://new.domain.com/new-page_url?newslug=with-different-slug (instead of http://new.domain.com/new-page_url/with-different-slug), then you can make the value of the URL argument (= the value of the newslug argument in this example) available as new variable to Rules, simply by using the Rules URL Argument module. Here is a quote about it (from its project page):


  ... provides two rules conditions based on URL arguments:
  
  
  check if a URL argument is present.
  compare the value of an URL argument.
  
  
  It also provides an action that makes the value of an URL argument available as new variable to Rules.


The newly provided Rules actions and conditions can be found under the "URL Argument" conditions and actions groups in the rule configuration interface.

For an example of how to use it, refer to Comment # 3 in issue # 1686360 which is about "Pass an amount to the url, ie; site/content/node-title?amount=10 or /node-title?=amount:10". Here is the relevant part of it:


  
  add a "Check if URL argument exist" condition to your rule and set the "ARGUMENT" value to "amount".
  add "Provide URL argument value" action to the rule and set "URL ARGUMENT NAME" value to "amount" and in the "Provided variables" section "ARGUMENT FROM URL - Variable name" to e.g. "amount_from_url".
  
  
  You will now be able to use the amount value as a variable "amount-from-url" for other actions in this rule.


PS: If you're not familiar (enough/yet) with Rules, checkout the 32 (!!!) great, and free, video tutorials Learn the Rules framework, organized in 7 chapters.