I know exactly the effect you're talking about.
If the console screen isn't moving relative to the player's eyes then the effect isn't visible, but if the screen is sliding up and down or moving nearer/further away from the eyes, then dark horizontal lines appear to scroll across the screen. The effect happens under both fluorescent and non-fluorescent environmental lighting as well as in the dark with a backlight or built-in frontlight.
I've experienced this effect on GBC, GBA, GBA SP, DSi, and DSi XL, all of which I fixed by adjusting the potentiometer on the main logic board. The issue hasn't occurred on DS and O3DS or any other non-Nintendo portable devices with screens. My N3DS XL upper screen is affected too and like most of the people in this thread, I checked hi-res photos of the N3DS XL logic board from iFixit but didn't see any potentiometers. I did the sideways screen test and found that both my DS and O3DS are using TN displays while my N3DS XL has an IPS display, unfortunately it's on a newer sysNAND and can't be further tested with a Gateway card. Hopefully someone could come up with a homebrew application that can fix this via Ninjhax.
Here's a summary of devices tested:
GameBoy (DMG-001): Not affected
GameBoy Pocket (MGB-001): Not affected
GameBoy Color (CGB-001): Affected, fixed via potentiometer adjustment
GameBoy Advance (AGB-001): Affected, fixed via potentiometer adjustment
GameBoy Advance SP (AGS-001): Affected, fixed via potentiometer adjustment
DS (NTR-001): Not affected, TN display
DSi (TWL-001): Affected, fixed via potentiometer adjustment
DSi XL (UTL-001): Affected, fixed via potentiometer adjustment (problem returns after a few days)
3DS (CTR-001): Not affected, TN display
New 3DS XL (RED-001): Affected, no potentiometer on logic board