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It's a given that with relatively small blocks the Bitcoin main chain would not be place where most transactions are done (obvious). It isn't a given that means transactions would then move to Monero or Dash (or LTC or DOGE). They could in theory move to Bitcoin side chains or Lightning or something else closer to the Bitcoin sphere.
That still leaves the issue of what happens if the Bitcoin main chain has hardly any transactions as the block rewards run out. A small but highly-secure chain that serves as a monetary anchor for other transactional systems rather than as a platform for transactions directly makes much more sense if the rewards don't diminish down to nothing, but there is really no way to get there with Bitcoin's fixed supply.
This is exactly the fundamental problem with Bitcoin and I would argue that Litecoin has exactly the same problem. Dodgcoin could save itself by adding an adaptive blocksize since they have already made the hard social covenant change of a tail emission. Dash is an unknown until we see exactly what Dash Evolution will be (supposedly in around 18 months), in its current state it also has the same problem. Bytecoin (premine / ninjamine) not withstanding also runs into this issue. Go down the market capitalization list and the first POW coin that has long term viability is Monero.
Edit: Anything built upon the Bitcoin main chain will fail if the Bitcoin main chain fails and this includes Lightning, side chains etc.