[SPOILERS]:
I took my 6 PCs from LMoP to HotDQ straight away. They were 4th level when finished with LMoP and I included a note in the dragon cultist encounter in Thundertree that was from Rezmir to Favric asking him to attept to gain allegiance with the green dragon and then report to cult agents in Greenest. This was a direct link to HotDQ and since the PCs elected, wisely, to tackle the dragon last (at 4th level), they pretty much transitioned from LMoP to HotDQ easily and probably wouldn't have even known they had started a new scenario (or that Thundertree wasn't the start of HotDQ unless I had told them.
As far as HotDQ, at 4th level then encounters in the first episode are not deadly like they are at 1st level, and the PCs could handle the whole episode without completely running out of resources. Nevertheless, running all of the episode 1 encounters, plus having the dragon be a bit more aggressive towards the PCs helps. Plus the DM can be liberal with the wandering encounters. All told, even a party of 4th level PCs running the encounters straight out of the box found themselves taxed overall and in a few sticky situations. Even the duel with Cyanwrath should have resulted in a defeat for the PC challenger except that I couldn't roll a hit for the poor guy to save my life.
One nice feature is that because the PCs were 4th level, were given a reason to go to Greenest, and had just defeated a dragon, they had no inclination at all to flee or prevaricate when I described the Huge blue dragon flying over the town.
The PCs are now well into episode 3 and are still 4th level, so the scenario is rapidly catching up to them.
In any event, I highly recommend a direct transition between the two scenarios. Although they are not designed to follow, they transition quite well.