In most cases it will not cause any issues with regards to SEO noting speed and performance are factors to consider with regards to your bounce rates.... In addition in some circumstances you may find that the new web server you are on has been blacklisted should it be on shared hosting due to activities of one or more websites on that server... This can also apply to email due to self hosting.. There are tools online whereby you can see the domains listed for a given web server, so it maybe worthwhile to check that as well as for IP blacklisting checks... One way would be to take that list and check that against G's search with checking that all sites are listed in the search index...
If you are moving a wordpress site then I recommend that you use the duplicator plugin as that way you can migrate the database, files/folders with a single file, upload it on the other end and install with a few settings/clicks.. Literally the whole thing can be done within under a minute... For other platforms you need to consider database exports as well as files/folders with a manual copy via FTP etc...
