Lucario's ground game is outclassed by Bowser's ground game, so Lucario is at a disadvantage on the ground once he can't spam Aura Sphere. Lucario's air game is much better than Bowser's, but this isn't much of a problem since Bowser's ground game is better than Lucario's aerial game, so Lucario must be careful. Lucario's approach options are mostly limited to aerials and grabbing since Bowser's Ftilt will out prioritize and out range virtually everything that Lucario has other than Fsmash and Aura Sphere. Bowser can get around Aura Sphere by shield dashing/walking and Infinite Jumping with his Side B.
Lucario has a chain grab on Bowser at low percents, but with proper DI Bowser should be able to escape before 30%, which is not as bad for Bowser as it is for other characters thanks to his heavy weight.
With good DI, Lucario should live to ~120% and Bowser should live to ~160%. Bowser also has a ground release chain grab on Lucario that lasts until Lucario jump breaks or Bowser stops grabbing. Ground releases also lead into inescapable Dtilt that kill ~100% on the ledge of most stages or Side B which is ~19% in addition to whatever pummel damage was accumulated.
Bowser gimps Lucario better than Lucario gimps Bowser. Bowser's Neutral B can gimp or rack up quite a bit of damage [if Lucario tries to DI through it] should Bowser come from below the stage or use his aerials if Lucario comes from the side: His Nair has a long lasting hit box that will hit Lucario if he tries to Up B through it and his multi hit Dair (it's ~11 hits total, I believe) is also long lasting and has a downward trajectory for the last hit. Bowser doesn't have a good answer to Lucario that is recovering high above the stage other than attempting a Uair which, can kill Lucario at ~70%.
Lucario can hit Bowser with Aura Spheres and aerials if Bowser comes in high or from the side, but Lucario doesn't have a real option against Bowser's recovery below the stage other than ledge hogging. Note: Bowser can use his Down B to quickly reach the ledge from above the stage. This can be beat out by properly spaced aerials and Aura Sphere, but it's fast, deals ~20% should you trade hits or miss the timing, and is very safe since it ledge cancels.
Bowser's out of shield options are terrific and can punish virtually anything that Lucario has. If inputted correctly, Up B out of shield has no vulnerable frames before the high priority hit box comes out (it clanks with Snake's Ftilt) Bowser can while using Up B once the hit box comes out, and he can use this to punish spot dodges, rolls, and resetting his position. Neutral B fires flames that are similar to Charizards, but they are wider and hence edge guard Extreme Speed better than Charizard's Neutral B. It is a good idea to DI away and, if the Bowser unwisely keeps up the flame (most good Bowsers will stop their flame once you DI to the end of their range), use Double Team.
Side B is a good move that deals good damage (~19%), can be used to Bowsercide, kills at high percents, and can be used with the Infinite Second Jump Recovery to constantly have a second jump while on stage without needing to land. Note: Bowser can kill with a Bowsercide and survive it on some map transitions such as the Help Ghost on Yoshi's Island or the transitions to a stage with water on Delfino. Down B has two hits when grounded: The rising hit which deals little damage, has almost no knock back growth, and leads into the aerial hit. The aerial Down B is the same as the second hit of the Down B. It deals good damage (20%) and the grounded Down B is one of Bowser's best kill moves and will kill ~80%. The grounded Down B can be air dodged if hit by the rising hit.
Even though most of his moves are kill moves, some Bowser's save their Dtilts as kill moves since it can easily be performed out of a ground release and has a gimpy trajectory that will hurt Lucario on maps that have sides large enough to allow horizontal gimping (if Bowser hits you with a Dtilt at the ledge at ~60%+, landing a Fair will put Lucario too far out to recover). Utilt has very high priority and range and kills about as well as Snake's as Utilt; don't underestimate it and be careful if approaching Bowser from behind, as the hit box covers above and behind him for a ~120 degree angle. Bowser's first jab hit combos into the second hit and both hits deal ~5% for ~10% total. The jab starts fast, and while the range is shorter than his Ftilt, it still rivals many other characters' Ftilts.
Fsmash is easy to avoid but shouldn't be underestimated, as it can definitely be landed. The Fsmash is made of 2 hits: When Bowser begins to move forward (his abdomen area is the hit box) and his head once he begins to slam forward. The head is the main concern for knock back and the tip of the attack (his horns) is a sour spot that has smaller, completely vertical knock back as opposed to the normally diagonal knock back. Usmash hits twice: Upward and then downward. The Usmash kills better than Utilt when sweet spotted, but on the ground Utilt is the bigger killer and doesn't need to be directly under Lucario. The sweet spot is the center spike on Bowser's back at the apex of his jump during the attack. Dsmash is the smash attack that can be expected, as it can punish spot dodges and rolls as well as incorrect recoveries. Dsmash can be DId out of depending on when and where in the Dsmash you were drawn in. The final of the seven hits hold the knock back. All three smashes kill at mid to high percents even with good DI: Fsmash kills at ~80%, sweet spotted Usmash kills at ~100%, and a full Dsmash (all 7 hits) kills at ~110%.
Bowser's aerials are telegraphed, but don't make the mistake that they are easily avoided because of it. Bowser can bait air dodges just like everyone else. Fair is a good move for gimping off stage due to being one of his faster aerials with decent forward range with decent damage. Bair also does decent damage with decent range and is also a kill move. Bair will outright kill at ~90% off the ledge, so be careful when he is jumping backward to you. Uair is a spectacular kill move that will kill Lucario virtually anywhere on most stages from ~80%+. Dair will not be seen very often and can be DId out of. Don't take it lightly, though; it deals 20%+ damage if you don't DI it and has a deceptive landing hit that knocks Lucario far enough back at all percents to make it safe. It also is for shield damage and should be watched out for after a shielded Fsmash. Nair has a long lasting hit box that is Bowser's body during the attack. It doesn't deal as much damage or knock back as his Fair and Bair, but it still works as a good edge guarding aerial.
Lucario does not want to take Bowser to Norfair as the only character that Bowser shouldn't take there is Meta Knight, as that is one Meta Knight's best stages. A good stage against Bowser would be Jungle Japes. The ceiling helps Lucario survive and the terrain is much better for Lucario than it is for Bowser, although it doesn't allow for as much Aura Sphere spamming. A good neutral map is Yoshi's island since Bowser can't gimp you very well thanks to the large area for wall clinging. Dream Cruise is also a good idea since it will keep Bowser in the air with Lucario, although Lucario doesn't really have good options of Aura Sphere. Make sure to ban Corneria, as the survivability should have Lucario living to ~90% and Bowser to ~140% and Bowser's Uair kills at early to mid percents (50%+) which is viable against Lucario when he above Bowser, particularly at the fin.
In my opinion the Bowser-Lucario match up is 40-60 in Lucario's favor.
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