Hi

Mid range the revs 'bog' down and then clear and the bike takes off - any suggestions please.

If I remove the air box cover to let more air in its fine - revs cleanly throughout the range to red line.

New or old filter element?  Old filter foam has been known to break down and restrict flow even though it looks fine.  I had this happen on my 1972 LT2 back around 74-75 so foam doesn't need to be "ancient"

Yamaha filter or aftermarket?  Uni filter is known to be more restrictive than original.

Modern "sticky" air filter oils or over-oiling (easy to do) can cause issues...  I prefer Twin Air foam filter oil...  has been around since the 70's and works great and easy to use...  virtually impossible to over-oil. 

I found a K&N filter for the DT360A I have, seems ok so far. Heavy grease applied at the ends for sealing.

Like MarkT said, don't over-oil.

“Old filter foam has been known to break down and restrict flow even though it looks fine.” 

Last time you mentioned that about old filter elements I checked all my filters. I wish I could post the picture of the one that disintegrated when I tried to remove it from the cage. It was like pulling cotton candy off the paper tube at a carnival. 

I fitted a new genuine filter and only very lightly oiled?

Did you get the filter from Yamaha or a "NOS" one from ebay or someplace?

What oil did you use?  The "air filter" oil sold by UNI, Maxima, etc. is often too thick/sticky. 

(I personally struggle to "lightly oil" a foam filter with any kind of regular engine oil.  Many racers used to use old premix to wash the filter and when the gas evaporates the theory is just enough oil is left in the foam.  Seems to work good.  Twin Air oil is similar but it's designed for filtering air.  It's like blue water out of the can.  I soak the filter and drain excess back into can.  Then let filter sit overnight to dry.  I started using Twin air oil in the 70's when I got my Husky and still use it today.)

Worn needle jet & needle in the metering part at the top of the emulsion tube.
Speck of old filter in the airbleed to the needle jet from the small hole in the mouth of the carb.
Video if pos would help diagnosing but on the face of it--it sounds rich lower to midrange i'm guessing.

I use Uni filter spray, Maxima filter oil or even reg 10w30...... on oem furries, Uni foam or aftermarket foam filters and it's only a problem if you over oil them. In my shop we always squeeze at much out as possible no matter what oil is used, the foam just needs an oily feel to it never wet. As I live in Calif, using premix to oil the filters is a no no since we are blessed with Ethanol fuels here (unless you wanna use race gas) and although it's interesting to see foam react to ethanol it's expensive.. lol

Just removed the air filter - it didn't appear to be over oiled but I thoroughly washed it through and dried it out.

Fitted it with no oil - the bike revs cleanly in each gear to the red line.

Should I just apply a very very little amount of oil or leave it?

run it without the filter around the block isn't going to hurt anything. Can't get any 'less' restrictive than that. Unless something's stuck in the airbox intake. Did see that once on something that sucked from under the seat & had left something under there. Had a seat collapse too on a road bike with similar results.