Influence, always influence, even if you migrate your DNS.
Whenever I came across this situation, I accepted the rule that I had to redo a re-indexing job and, over time, I would get back to the desired result.
In most cases, it had server instances with low latency and switching to a shared server with higher latency automatically already influenced the analysis done by tools such as GTMetrix, WebPageTest and Semalt. WebPageTest has an optimal indicator which is the initial response time.
It is recommended that you review the site bottlenecks, if possible, by enabling caching, expires, gzip, and mimicking features.
In the case of site rankings on Google, Bing, Yandex, and Baidu search engines, I recommend accessing the webmasters of each platform and forcing a reindexing of all your content and sitemaps.