Doug 98 R34 GTT

Hoping someone can shed some light on this... All 1 - 5 shifting is smooth, all 2 - 4 down shifting is smooth EXCEPT when I'm in 5th, go to neutral, then down shift to 4th (speed matches 3rd/4th, could've possibly been a bit slow for 4th). It grinds and won't go in, almost feels like as if the clutch is not pressed in. I typically don't go to neutral then 4th, but today I did for whatever reason and noticed it. It will happen every time, but if I go straight from 5th to 4th, zero grind, very smooth. Super odd. I haven't tried double-clutching to see if that helps, but tbh, I'll just never go to neutral, just wondering what would cause it to grind only in that scenario.

Thank you both, much appreciated. Yes, 6k miles, not a typo lol. The underside, engine bay, engine, trunk, interior, etc., look great and super clean, has all the splash guards too. Not sure if you can expand the photos, but the dash pic shows the ODO. She looks brand new, inside and out, even has the original Nissan TV/DVD/Nav head unit; still thinks it's in Tokyo haha! I used to own a '94 MKIV Supra, sadly sold it years ago, but wow I love the Skyline so much! The R34 gets SO much attention here since they just became legal to import last year.

New R34 owned here (from the US) and found a gem, '98 GTT with ~6k miles, super clean & completely stock. When shifting there's no grinding, shifts smooth, etc. The concern I have are 2 things: 1) Only in 2nd and 3rd... when maintaining a slower speed (in traffic as an example) as I let off/on the gas, the shifter will slightly move and vibrate. If the clutch is engaged, none of this happens. It's not bad, but enough to feel like something isn't necessarily right. 2) This mainly happen in 4th or 5th... the shifter slightly bounces up and down, almost like it's riding on unbalanced tires (and I do think the tires are unbalanced as I get a little of that bounce feel at higher speeds, which is maybe why it only happens in 4th/5th). Lastly, and sorry for this noob question but I want to work on this car myself, can someone teach me some on these manual transmissions? Like servicing them, best products to use, etc.; would be greatly appreciated. Thanks!!