Free Agent - NA
This tierlist is my personal ranking of who has the best mobility in the game. I give detailed descriptions on what the mobility is and what purposes it can be used for. (Card Loadouts and Talents will not be included for a champion unless stated.)
ss
Androxus and Evie are quite possible have the best movement options out of everyone. Androxus' "Nether Step" is a dash that lets you dash in any direction you're facing, including up. for a short while you can dash a second time and once again for a third. The timing window for his "Nether Step" is also big. He also has a passive that lets him hover over the fight or over cliffs, letting him wait for "Nether Step" to come off cooldown. His ultimate "Accursed Arm" even lets him fly over the battle. Evie's "Blink" and "Soar" are quite literally the best mobility in the game. With "Blink" she can instantly teleport around and with "Soar", she can fly around anywhere. When mastered correctly, she can travel miles in an instant and can literally fly circles around her foes.
s+
Maeve and Vatu come in 2nd place in terms of best mobility. Maeve has both her "Pounce" and "Prowl" which give her a dash and an increased movement speed respectfully. She also has a double jump that is enhanced with "Prowl". Vatu's "Dash" is very useful with multiple charges and quick cooldowns, however it uses movement inputs to dash which can be either a blessing to a veteran/pro and a disaster with a beginner. His "Ambush" is also hard for newer players to master as it works on enemies, throwing them into the danger, but if used right, can cause disaster to the enemies. Vatu's ultimate "Nightfall" is also a decent dash attack if either distance or height is required.
s
These champions have very good mobility. Buck and Octavia's "Heroic Leap" and "Commanding Leap" act the same way with Buck's having more air control and Octavia's letting you leap in any direction. Both give great distance and probably the greatest option to high ground access. Grover and Vora's "Vine" and "Tendril" are also the exact same with Vora's having more control after using. Both are limited by range, but both let you arrive to the scene faster or let you escape quicker. Imani's "Frostfire Glide" is a flight ability similar to Evie's "Soar", but it's weaker when it comes to gaining height or turning. Other than those 2 issues, it's one of the best mobilities in the game. Ruckus' "Advance" makes him one of the best Frontline with his ability to quickly dash in and out of combat and, if using the "Aerial Assault" talent, lets him even gain the high ground if needed.
s-
These champions have decent mobility and usually gain benefits from doing so. Cassie's "Dodge Roll" is a great get-away card as it has great speed and good distance. Corvus' "Projection" makes him one of the best Supports, mobility wise. It has decent range and acts as a teleport, and if used right can even stun enemies. His ultimate "Entropic Breach" also lets Corvus fly around for a short time. Drogoz is possible the best champion for aerial mobility as his "Thrust" and "Booster" abilities let him stay in the air for a while. His ultimate "Dragon Punch" also lets him fly super fast around the map for a short time. Raum's "Juggernaut" is a great mobility for a Frontline to have and lets him become a reckoning force on the battlefield. Koga's entire kit is mostly around mobility. His "Agility" increases his movement, "Shadow Step" lets him dash while invincible, "Skewer is a great dash attack, and he has passive wall-climbing. However he is weakened by the fact that most of his abilities require energy rather than having cooldowns.
a+
These champions have okay mobility, but are mostly used for other things. Ash's "Shoulder Bash" has great range for a dash, being one of the longest, and great utility as a knockback attack against enemies. Her ultimate "Assert Dominance" also lets her jump into the air giving her access to the high ground if possible. Barik's "Rocket Boots" is a great tool for dashing in or out of danger, along with the turrets and shield, he's not going to be easy to catch. Bomb King's "Poppy Bomb" is great for sending you flying, but can be used more effectively by knocking enemies away from objectives. Io's "Lunar Leap" is a great get-away card and with it's range, can travel far as well as retreat back to your team, if needed. Moji's "Scamper" is a decent burst of speed and can be used to confuse enemies if to close. Pip's "Weightless" is easy to use, press the button and you gain increased jump height and speed for a duration letting you get out as fast as you can get in. Talus' "Rune of Travel" lets him have a safe location to travel to if he's injured or in a bad spot. His "Blitz Upper" is also a great dash to use but is more effective as a dash attack. His ultimate "True Power" can also be used to get to the enemies quicker if your team is at a standstill. Tiberius' "Crouching Tigron" is a great multi-direction dash and is easy to use. Downside is its long cooldown. (Longest out of all of the most available ones in the game.) His ultimate "Blade Dance" can also be used for movement with dashes and upward spin attacks. Willow's "Flutter" is great for covering distance as well as gaining high ground, if needed. Her ultimate, "Fae Flight" is also a great mobility option as you can soar through the sky.
a
Most of these champions have mobility options, but can also be used as attacks. Furia's "Wings of Wrath" can relocate her behind her team while also damaging enemies with fireballs. Khan's "Commander's Grab" is great for stunning enemies or throwing them to your team to finish off. Makoa's "Shell Spin" is a decent dash with decent damage as well letting him use it with his "Shell Shield" for more utility. Terminus' "Shatterfall" is great for stunning large groups and giving him some high ground if possible. Zhin's "Whirl" is great for damaging lines of enemies and getting behind them. It's also a decent dash for travel, but should be used as an attack. Dredge's "Shortcut", Jenos' "Stellar Wind", Kinessa's "Transporter", Sha-Lin's "Withdraw" and Viktor's "Hustle" are all great movement options especially Dredge's "Shortcut" letting him possibly move from one side of the map to the other in an instant.
a-
These champions have mobility but it's not that great. Fernando's "Charge" is not the best dash in the game, but he can still use his shield while using it, letting his team push forward if needed. Lex and Lian's "Combat Slide" and "Grace" respectfully both act the same way, as a small relocation ability. Saati's "Blast Back" is an okay ability while running the "Window of Opportunity" talent. Other than that, it's not all that great. Yagorath's mobility comes from her "Travel Form" where she curls up and moves similar to a mount. Unlike a mount however, her movement is a little crazy at times. Ying's "Dimensional Link" is good to get out of cover, if you have a distant illusion outside of combat. If not, you run into some issues, but it's a great ability to confuse your enemies.
b
Mal'Damba and Rei aren't very good when it comes to mobility. Mal'Damba has his "Slither", but you can't do anything while it's active and it makes you an easy target once it's done. Rei's "Envelop" give her and her linked ally increased movement speed, but its mostly used for the damage reduction that it provides.
c
These champions don't have mobility, but the got options. Atlas' mobility comes from he's "Second Chance". He rewinds back to his past position which, depending on location, can actually throw you back into danger. Seris is higher on this list than Grohk because her "Shadow Travel" lasts longer and give her extra benefits. Inara has no mobility, but with her "Summit" Card, she can launch herself with her "Impasse" walls. Not a good mobility, but a mobility nonetheless.
d
These champions either have little mobility or none at all. Grohk's "Ghost Walk" is too short to be effective and Skye's "mobility" comes from her cards. The other champions have no mobility at all.