Does the "follow" instruction give an aircraft clearance to turn onto a taxiway and/or cross a runway?
The controller is required to give explicit clearance for crossing a runway.  &quot;Follow&quot; is not clearance to cross a runway.
From the ATC phraseology:

7-3-2
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
applicable.
EXAMPLE−
“Follow (traffic), cross Runway Two−Seven Right.”
or
“Follow (traffic), cross Runway Two Seven−Right, hold short Runway
Two−Seven Left.”

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
route.

That said, I would not expect the only instruction from the controller to be &quot;follow (traffic)&quot;.  I would expect to be cleared to a specific destination as well.

&quot;N456, taxi to 36.  Follow N123&quot;