I've done this anecdotally (i.e. not with a huge quantitative sample size, but looking at specific "hot" tech companies and the tech stacks they end up using) and it seems to work, but is subject to certain gotchas. Data points:
EBay (1995) - Perl, rewritten in C++ in 1997, rewritten in Java in 2002. Google (1998) - C++. Del.icio.us (2003) - Perl. GMail (2002) - Perl, rewritten in Java circa 2002. LiveJournal (2002) - Perl. Flickr (2004) - PHP. Facebook (2004) - PHP. YouTube (2005) - Python. Reddit (2005) - Lisp, rewritten in Python in 2006. AirBnB (2006) - Ruby on Rails. GitHub (2007) - Ruby on Rails. Dropbox (2007) - Python. Hacker News (2007) - Arc, YC internal tools (c. 2012) are written in Ruby on Rails. Twitter (2007) - Ruby on Rails, rewritten 2010 in Java/Scala. Uber (2010) - Node.js, supposedly this was a rewrite.
There are two big caveats:
Sometimes a big competitor comes along right when a language is undergoing a massive rewrite - for example, it looked like Perl would own the Internet around 2003, but just a couple years later we got Rails and Django and Perl stagnated with Perl 6, and so Perl jobs are nowhere near as hot as they would otherwise be. Similarly, Python was poised to take over much of the non-web server space in 2008, but in 2009 Node.js came out just as Python was losing steam in the Python 3 transition. Python still remains pretty hot (it's been helped by its use in data science), but it may've lost the startup server market to Node.
The other big anomaly is that there was a lot of attention around Haskell and Erlang from 2005 - 2008, and yet these are still niche languages. Erlang had one big success with Whatsapp and some smaller ones with Facebook chat and CouchDB, but didn't really become mainstream. My hypothesis there is that because both of these were old languages that were rediscovered, they were built with some assumptions (eg. strings are lists, or funky POSIX interfaces) that didn't fit modern startup development, and so they couldn't gain critical mass. But then, both Python and Ruby were old languages that were rediscovered; perhaps it was the sudden resurgence of Linux (which both Python and Ruby were steeped in) along with the web that carried them forwards.