So for portability, low light, speed for sports photography and action. I would want a Sony A9. I would settle for the Sony A7r II as well. There are many things that make the camera stand out the main ones are the eye focusing. Sony has created a camera that can focus in on the eyes of your subject so your eyes will always be in focus, don't worry if you want to focus on other stuff just touch the screen where you want the focus. It also has dual media slots so you don't lose anything when a card acts up.
A battery for the Sony A9 can last a day, shooting around 7000 images or more.
If you have other lenses there are adapters for them to use.
Doing Action Photography or sports you can fire off 20 frames per second up to 240 RAW images in the buffer. That's going to get you through the action.
As for off camera lighting: For portability and action and just easy to use. Rotolight Anova Pro LED lights. They have a really nice light and a 50 degree spread. Soft light. Fixed 5600K +- 100K color temperature, Correction filters to make it 6300K, 4400K or 3200K. Flicker free continuous output. Flash Sync/ remote triggering via 3.5mm PC mini-jack. V-Lock battery or A/C power. TVMP bracket. This light in High speed sync has no recycle time. I think it goes 11fps.
If your setting up a studio then you may want to go with something else. I would just use more. The controllers for these can do over 200 lights at once.
Price is high for both, I feel it is worth it once I save the money.