In case of a change in the URL, it is compulsory to establish redirections between the old pages and the new ones.
Indeed, during a redesign/migration, Google will use these 301 redirects (permanent redirection) to allocate to your new pages the ranking history of old pages.
This period of transition can last between 1 to 4 weeks - in order for Google to index your new URLs instead of your old ones. This period of transition will usually cause a slight loss of organic traffic, in order for Google to appreciate the quality of your new pages.
It is usually a good time to delete obsolete pages you did not know about or even sub-domains in order to reorientate Googlebot on the right pages.
Migration plan
Your SEO team will provide a migration plan: a document that indicates the correspondance between old/new pages in order to establish the useful 301 redirects to make. The idea of this document is to identify a maximum number of URLs and URL formats to ensure that no search engine will waste time on the old version of the website.
For an Apache server (the most popular one) using the rewriting module "mod_rewrite", the best way to redirect a page towards another one is to use the .htaccess file located at the root of your website.
In order to redirect a URL to another, place the following line after the instructions Options +FollowSymLinks and RewriteEngine on, then customize it as much as possible for each redirection:
Redirect 301 /old-page.htm http://www.mysite.com/new-page.html
It is also possible to redirect a file and all pages using that file to its new name:
RewriteRule ^/docs/(.+) http://new.mysite.com/docs/$1 [R,L]
==> More instructions on the official literature for developers here: https://httpd.apache.org/docs/trunk/en/rewrite/intro.html and https://httpd.apache.org/docs/trunk/en/mod/mod_rewrite.html