In a previous game, one of my players played a Bard that got up to level 5. That player wants to bring him back for a few sessions when he gets his current Artificer up to level 5 (which will probably be around 2 sessions away). Both campaigns are set in the same world, the new one a few years after the end of the previous one, so it makes canonical sense. The Bard character, however, doesn't know any member of the new party (though the players do).

Normally I have a rule (I don't know if it's official or a house rule), I've just always played it this way) where if your character dies, you create a new character that's 1 level lower. I've never had to deal with a player swapping out a character "just because".

Are there any rules for players swapping out player characters of equal level?

There are no rules for this, but the DMG does have a few suggestions about "backup characters"

The DMG doesn't include any hard-and-fast rules that address this issue in detail (and neither does Xanathar's Guide to Everything).

However, it does briefly touch on the idea on a tangentially related section of DMG p. 236 titled "Small Groups". The section is primarily focused on campaigns with a small group of players, proposing that each player (who feels comfortable with it) control multiple characters, or that the DM fill out the group with NPC followers. However, it does add this suggestion at the end:

Multiple characters can be a good idea in a game that features nonstop peril and a high rate of character death. If your group agrees to the premise, have each player keep one or two additional characters on hand, ready to jump in whenever the current character dies. Each time the main character gains a level, the backup characters do as well.

This guidance suggests that if the player's character dies, the backup character would be the same level as their original.

However, there's another mention of "backup characters" on DMG p. 92, under "Low-Level Followers":

Your campaign might allow player characters to take on lower-level NPCs as followers. For example, a paladin might have a 1st-level paladin as a squire, a wizard might accept a 2nd-level wizard as an apprentice, a cleric might choose (or be assigned) a 3rd-level cleric as an acolyte, and a bard might take on a 4th-level bard as an understudy.

One advantage of allowing lower-level characters to join the party is that players have backup characters if their main characters take time off, retire, or die. One disadvantage is that you and your players have more party members to account for.

Since lower-level NPC party members receive equal party shares of XP, they will gain levels more quickly than the adventurers (the benefit of studying under such experienced masters), and might eventually catch up to them. It also means the adventurers’ advancement is slowed somewhat, as they must share their XP with an NPC shouldering only part of the adventuring burden.

This suggests that a player's backup character might not necessarily be the same level as their original character. (It also assumes that the DM built the NPC as they would build a player character rather than as they would normally build an NPC.)

It's up to the DM

As the DM, it's ultimately up to you to decide whether to let the player swap their character out for another character of equal level. You can also talk to the players to see how they feel about it.

Remember that the point of D&D is to have fun, so the goal probably shouldn't be to force a player to continue playing a character they're not enjoying... But on the other hand, it can be difficult to be immersed in a campaign or tell a coherent story - especially a character-focused one - if players swap out characters too often. (And, of course, figuring out how to explain the character change in-universe is its own problem.)

Ultimately, this will need to be a conversation between the DM and the players. Do what feels right to you and your players.

To my knowledge there are no rules in 5e that govern "swapping out" characters or to returning one to the same game, let alone a different campaign. Like almost anything in 5th edition it's entirely a matter of DM fiat how any of this proceeds, if at all. There are some suggestions - not rules - relating to multiple characters but they don't appear to cover what you're talking about.

That said, many campaigns do have players retiring one character, playing a new one and then having an old character show up for some specific reason. Most commonly a retired character is an NPC, not a second PC (i.e. it would no longer be under the original player's control) but this (again) is entirely at the discretion of the DM. The 5e rules (as defined in the PHB and the DMG) don't speak to this, it's a matter for negotiation between you and the player (and possibly the other players - I might find it okay as a DM but I still wouldn't do it if the remaining players had any substantive objections).