For stage selection I wouldn't take Falcon to YS or DL.
YS he can take up a lot of space with his aerials, and often juggle you on platforms with upairs, and the walls and Randle (the cloud) really aid his recovery. I find the small blast zones help him more than they do you, as his knee kills at incredibly low %s and are risky to parry/easy to land. Also, his limited recovery is better on this level and he can almost always up-b back to stage and mix up with landing on the stage or going for ledge, and can really trick up your edge guarding, but that is just my experience. Don't recommend vs. Falcon unless you really like this stage or find you have some other advantage here.
DL gives Falcon a lot of room to move around, and as Falcon excels at tech chases and reads, it gives him lots of space to do the things he is good at. He can also cover the higher platforms with Uair pretty well, and escape your pressure due to the size of the stage coupled with the high platforms that he can waveland onto out of most of your moves' reach. He is also able to recover a lot easier with Falcon Kick due to his weight and the wide blast zones. All in all, I don't like this stage too much vs. Falcon.
On FD Falcon has a lot of movement and free space for combos, and his combo game on Yoshi is pretty ridiculous, but if you have a strong combo game yourself, this can be a good stage for you. The lack of platforms really restricts Yoshi's movement as well, and it's hard to mix up your approaches on FD. I'd recommend mixing up your DJC height by using fullhops to DJ Bair/fair to come down onto him with, as well as lower approaches with Nair and Bair. Falcon also struggles with approaching safely and unpredictably somewhat, so its pretty 50/50, or even in your advantage here if you can stuff/Parry his approaches well and get high damage combos, and finish your edge guards often. Pick this stage based on how your opponent plays and how well you can combo.
Battlefield is definitely the most neutral in this matchup. This is a good stage to feel out your opponent and determine if you should take them to any other stage (if you lose here), or determine what stage you should ban (if you win here). I'd recommend this stage as a starting pick or CP, as it can be good in general. You can reach through the platforms with a lot of your moves, which means Falcon won't feel safe being on a platform. You can really utilize the top platform for your waveland approaches, and lightshielding can be used as a (somewhat) safe defensive option or as a mixup into a shield drop uair to start a combo. Edge guarding Falcon with eggs when he is recovering from below you can be really useful, as often times you can get him caught under the stage if he is trying to reverse up-b grab the ledge. The lack of a wall means there is less room for error for Falcon to grab ledge, so if he miss-spaces his Up-b you can get him back off stage at a sharp angle using down smash, and it is often hard for him to recover from it. On the other hand, Falcon has a lot of platform control as well with his quick wavelands and his drop-through Uairs.
Pokemon Stadium is a stage I consider in most MUs as a CP. Yoshi's combo game here can be amazing, and the low platforms let you harass opponents with Uair if they get stuck in shield on them, and they allow nice movement mixups. Shielding on platforms can turn something generally bad vs. Falcon, into something beneficial. If you can turn shield dropping into an aerial like Nair/Uair/ waveland fair, it can start a tech chase or combo opportunity in your favour. Even if you're stuck on platform and must lightshield, it can get you out of a dangerous situation by baiting a strong move, sliding you to ledge to spam ECEs and get back safely. Yoshi can find kills off the top here with Uair and Usmash, and the transformations really benefit your ability to egg camp Falcon. For Falcon its the same situation here as on FD or DL, but he can't use his platform movement as well as he can on other stages, and often fears getting stuck on a platform. The good here outweigh the bad for Yoshi, in my eyes.
As for where to CP to, if they don't ban FoD, i'd definitely take them there. Most Falcons absolutely HATE this stage. The moving platforms really hinder Falcon's movement, and often cause him to miss-space aerials. The moving platforms also allow Yoshi's waveland approaches to differ all the time and removes predictability from them. The larger blastzones are also better for Yoshi I find. If they don't ban it (which I doubt they won't), and you're comfortable here, then go here.
Please let me know what you think of this and I will update it as you point things out to me. Let's get these threads thriving!
I also kept my thoughts on PS there, just because I like that stage in general.
