When moving from http to https you will see http drop and https rise until all the 301 and 410 are integrated into https (google webmaster tools treats each protocol as different sites)
As how to proceed, in my experience you just add the https version as a new property, set up the redirects in your server: use 302 (found) until you are sure the urls are right then change to 301 (moved permanently).
Replace all javascript and css calls to be protocol agnostic:
<script src="http://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
becomes
<script src="//ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
Fine tune the redirection rules to avoid multiple redirections (for a couple months you will have one extra) ie:
from http to https and then to https with www.
for http://yourdomain.com it could be two redirects :
http to https <- first redirecthttps without www. to https://www.yourdomain.com <- second redirect
that could be just one redirect tweaking the rules:
http with and without www. send them to https://www.yourdomain.com <- one redirecthttps without www. send them to https://www.yourdomain.com <- one redirect
To "speed up" the transition and cleanup the index it's advisable to submit a sitemap and if it's not already there add it to the end of robots.txt too something like:
User-agent: *
Disallow: /demos/
Sitemap: https://www.yourdomain.com/sitemap_index.xml
Once sitemap is processed and robots.txt checked do a fetch as google
Depending on your domain as google sees it, this transition time could last between 1 and 5 weeks, and after 4 to 6 months all should be normal again in the https property (http will still get residual hits)
