Overwatch

Does the target of my Overwatch attack have to move to lightly obscured areas or start its turn there for me to have disadvantage?

It would technically be whether the target is in a lightly obscured area (such as dim light, patchy fog, or moderate foliage) at the instant they have completed 10 feet of movement within your line of sight while within 60 feet of you, however that minutiae would ultimately be up to the DM at your table, as with everything. It would be fair enough to rule that if the creature starts its turn OR moves into a lightly obscured area, the ranged opportunity attack has disadvantage.

Unless otherwise stated, this ranged opportunity attack still follows the rule of normal opportunity attacks, including being negated by the Disengage action or if someone or something moves the creatures without the use of their movement, such as being pushed or falling due to gravity.

It also still follows the rules of your ranged weapon, so if the ranged attack is outside of the weapon's normal range and into its long range, the ranged attack has disadvantage anyway.

This is one of my strongest homebrew feats. Speak to your DM about its usage, as it is not suitable for all campaign settings.

Thanks! My DM seems cool with it, so I thought I'd consult you. 

Awesome, I'm glad to hear that! I wish you luck and cool trickshots lol

I'd imagine this being pretty fun with the gunslingers winging and pushing shots, and if a ranger had a loaded spell (like hail of thorns) from their previous bonus action that would come into effect on an enemies turn, which is pretty unique! With the sentinel feat you could also synergize to do some pretty effective control of the movement of a single enemy creature where movement is a big part of their kit, like a mounted knight or a griffin. Overall pretty cool!

That's a little bit up to your DM there. In simplified 5e-style rules, it would be if the creature started its turn in a light obscured area (such as shadow, dim light, patchy fog, or moderate foliage).

However, if the DM was to be more precise, it would be the moment you fire at it, which is when it has moved more than 10 feet from its starting position while remaining within your 60 foot range. So, if the creature is in a lightly obscured area after moving 10 feet from its starting.

As a DM and amicable player, I would lean towards imposing disadvantage on such ranged opportunity attacks more often than not, because this feat is very strong. This is likely my strongest homebrew feat. Constant ranged opportunity attacks with little provocation is really, really good. If I used this as a player, I wouldn't recommend trying to rules lawyer to get non-disadvantaged shots when there was any argument for disadvantage. The majority of the time, there will be no lightly obscured areas anyway so it won't always come up at all.