Though Melty Blood has a reputation as a very air-based game, your ground movement is just as important as anything else.
The most basic movement available to you in MBTL. Walking allows you to slowly close the distance (or create distance) with little commitment, and can be cancelled into any other action, including blocking. This makes it useful for baiting and whiff-punishing moves.
Forward Dashing, usually shortened to just Dashing, moves your character forward a set distance very quickly compared to walking. You cannot block during dash. You can cancel both the dash and the braking animation into anything else after a short startup period, but you'll have to wait until the brake animation is over until you can Forward Dash or Walk again.
An extension of Forward Dashing. All of the same rules apply, running is just an extension of the distance your dash goes. If you wish to stop, you can press at any point during the dash and you will enter a braking animation. You cannot block during run, nor on the first frame of the braking animation.
Back Dash, sometimes called a Backstep, makes your character retreat with a large step backward. It has invincibility on startup and goes very far, allowing you to evade enemy attacks and escape pressure. Backdashing carries a huge risk though, it has a long recovery period where you can't block, allowing enemies with long reach or high speed to punish it on reaction. Don't be predictable.
You gain access to jump normals and air options when you jump. You can't block most grounded attacks while in the air, but projectiles and other air attacks are fine. Neutral Jump in Melty Blood is unique, you can slightly drift while in the air by pressing or , this allows you to microspace while in the air, which is a very important component to Okizeme in this game. You can't drift with forward or back jump, but the rest is the same. Some characters jump higher than others.
Very often called a Super Jump, this is a jump that goes much further and faster than a standard jump. You can Super Jump with or but in Type Lumina there is only a forward Super Jump, so both inputs result in the same thing. You have full air options out of a High Jump. Each character's High Jump has approximately the same height and speed.
Unlike a lot of other fighting games, your movement options in the air aren't linked. This means universally everyone gets 1 Double Jump and 1 Airdash, and these resources are spent separately, allowing combinations like Airdash > Double Jump or Double Jump > Airdash. Another thing to note is Moon Drive grants you more options in the air.
When you're in the air, you can double jump. This is functionally the same thing as jumping on the ground, but you only get to do it once before landing. You can still drift off a Neutral Double Jump done with . Aoko can do this twice.
In Moon Drive, you gain an extra air jump, allowing you to triple jump. The extra jump is quite small, so it's usually better for baiting anti airs or squeezing in extra overheads than for moving around. Additionally, it uses up one of your airdashes if you still have any left.
Double High Jump
A Super Jump in the air, used by some characters to get in for an overhead after an Air Throw. In Moon Drive, the extra air jump can't be turned into a Double High Jump.
Forward Airdash propels your character forward, the distance and speed of which is character specific. You're going to be using this to approach with overheads, and to do double overheads with airdash cancellable moves in the corner.
Air backdashes are much shorter and generally used for evasion, despite having no invincibility, they are good for getting out of the way if you think your opponent is going to anti-air. In addition, they're used on Okizeme to delay an overhead. Also see Instant Air (Back)Dash.
In Moon Drive you gain an additional airdash. The one restriction is that you can't airdash in the same direction consecutively, that is, you can't go the same direction twice unless you perform some other action inbetween (e.g. jumping or using an air normal). The unique exception is Aoko, who just gains a third airdash without any restrictions.