If the site is staying on the same server not much to do, no issues in principle for search engines with a change of design.
Better as far as possible to 301 redirect the dead URLs to a mix of their nearest equivalent URLs, rather than all to the home page. Also worth putting up a fresh xml sitemap with just the new URLs, not the redirected URLs. Google will pick that up but you can submit the fresh map via your normal route e.g. Webmaster Tools if you wish.
Allow a little time, see how the site settles down. If "slim line" means significantly less content, that could have an effect. A percentage of redirects and possibly structure change could also see a shift in search, for better or worse.
Interesting story, a well known site (about 30 million pages now) once took down a few pages without properly redirecting them, including a hub page with 4 links to main sections. They spent a long time trying to work out their Google "penalty" until they finally found out about the self inflicted wound. An awful lot of dollars blown because someone didn't think about impact, or suitable redirects.
Not so for most of us, you may see little effect but hard to comment without knowing more about the site. The change to responsive design sounds good, little option for most sites who don't want to lose visitors. Sure you have already but if not, check the site in old browsers. Better to create a few fallbacks than lose as many people as you gain.