I've done a fair amount of adventure converting this year, to and from Pathfinder First Edition, Pathfinder Second Edition, Starfinder, and Dungeons and Dragons. I thought an overview about how I approach an adventure conversion would be helpful. I've broken this into nine steps. I'll use "native" for the original rules set and adventure, and "target" for the new rules set and adventure you're producing.

First, read the whole thing. Mark it as you go along for strange things that were expressly designed to work within the native rule set (like occult rituals for Pathfinder 1E, or skill challenges for D&D 4E). But make sure you have a sense of how the whole adventure plays out. Keeping the same "feel" is probably the most important thing you'll be doing.

Second, make easy swaps. If an owlbear is CR 4 or Level 4 or whatever in the target rules, just change the reference to the target rules' bestiary/monster manual/etc. Adjust the number of them as necessary for the encounter-building math of the target system to match the encounter-building math of the native system (you might need one or two more or fewer monsters). Note that traps sometimes have some easy swaps, too, as most systems have staples like pit traps.

Third, make hard swaps. If your native adventure has monsters the target system doesn't have (or has them at a weirdly inconsistent level, like how specters in D&D are much lower powered than spectres in Pathfinder), find something else that's close in function and role. For my specter example, maybe use wraiths instead. You might also have to look at the space in which the monster fights and see whether you need to swap--hill giants in D&D are much larger than those in Pathfinder, for example, and so might not easily fit in the same spaces. As before, adjust the number to ensure the encounter-building math of the target system matches the native system. 

Fourth, reimagine, then swap. If you couldn't make an easy or hard swap for an encounter, maybe some creative reimagining can let you do so. Maybe instead of being jumped by jungle tigers, the heroes are jumped by giant spiders instead. This requires a little tweaking to make sure the environment matches the monster (removing evidence of past kills and replacing it with spiderwebs or desiccated corpses, for example). Then, number adjusting as before! I've found that most traps conversions fit into this space--it sometimes rarely matters if the trap is a swinging scythe or volley of spears, so long as the environment appropriately reflects it.

Fifth, do the hard monster work. Here is the most time-consuming part! This is where a specific monster is load-bearing in the adventure, so you can't easily reimagine it, and you can't make a swap. You'll just have to build the monster from scratch in the target game system, and include its stats. Note that this is almost always the case for NPCs, unless the target system happens to have ready-built NPCs you can refer to (as 5th edition D&D does). If it does, use those as much as possible. Although this point is primarily about monsters, here is also where you have to rebuild load-bearing traps as well.

I want to drop a note here about reskinning. When you rebuild stats in the target system, see what types of creatures the target system has, and port over stats that make sense. Numbers alone often stand independently of the monsters; if you need a huge boar, for example, maybe look at the gorgon (another tough quadruped) and strip out its petrification breath. The stats otherwise work, and you can copy them with only some slight tweaking and that will save you some time. When carefully done no one will realize you got your stats from there.

Sixth, tackle the hard rules parts. If you have something that's rules-intensive in the native adventure, you may have to recreate something similar in the target rules system. This would be the case with something like a verbal duel, skill challenge, or complicated occult ritual. Often, you can translate over the rules elements into the target system and it will run fine. But don't be afraid to just cut it entirely--a hard journey with several Acrobatics and Athletics checks across a mountain range might be best cut down to a few Constitution checks to keep the story moving.

Seventh, review for rules terms. Here is where you change Diplomacy checks into Persuasion checks or Interaction checks or Charisma (Persuasion) checks, or whatever is right for the new system. Be absolutely sure you check the DCs as well--a target number that's reasonable for players in the native system might be practically impossible (or hand-wavingly easy) in the target system. Because you've already read the entire adventure waaaay back in the first step, you know whether the task should be easyish, moderateish, or hardish, and set the number in the target adventure accordingly. 

Eighth, balance the treasure. Treasure is something that changes pretty dramatically between rules systems, so make sure the value of the treasure in the native adventure is about the same as the value of the treasure in the target system. As you've been doing all along, look for thematic similarities: if a genie is carrying a magic scimitar, don't turn it into a magic warhammer. (But keep in mind that reskinning isn't just for monsters--it also works for magic items!)

Last, read the whole thing. When you go from the very first word to the very last word of the target adventure, it shouldn't have any leftover items from the native adventure (if so, fix them!). It might not read exactly like an adventure written for the target rules, but that's probably fine. The two things you want to ask yourself as you go through this last step are these: (1) Could someone with a working knowledge of the target rules system but NO knowledge of the native rules system run this? and (2) Does this adventure keep the same "feel" as the native adventure I read back in the first step?

And now you've got a converted adventure!