The controller is required to give explicit clearance for crossing a runway. "Follow" is not clearance to cross a runway.
From the ATC phraseology:
d. When an aircraft/vehicle is instructed to “follow” traffic and
requires a runway crossing, issue a runway crossing clearance in
addition to the follow instructions and/or hold short instructions, as
“Follow (traffic), cross Runway Two−Seven Right.”
“Follow (traffic), cross Runway Two Seven−Right, hold short Runway
The AIM doesn't spell this out as explicitly. AIM 4-3-18-5
- A clearance must be obtained prior to crossing any runway. ATC will issue an explicit clearance for all runway crossings.
The explicit requirement for runway crossings is somewhat new (2010 I think) and was the more interesting part. I neglected the question about crossing taxiways:
Does this [follow...] instruction mean N456 can turn right at Kilo without further ATC instruction?
Yes. Also from 7-3-2
a. Note 1. The absence of holding instructions authorizes an
aircraft/vehicle to cross all taxiways that intersect the taxi
That said, I would not expect the only instruction from the controller to be "follow (traffic)". I would expect to be cleared to a specific destination as well.
"N456, taxi to 36. Follow N123"