Making this mostly for personal reference but it may be useful to others:
Leveled up in 4x27: Murray (gems), Tyranny (knife)
Leveled up in 4x28: Thaisha (Thjazi's grave/Calling the Corners)*, Wick (return of the prodigal son), Azune (Einfasen meeting), Vaelus (conversation with Hannan)
Leveled up in 4x29: Occtis (touching the veil), Hal (KoTher'ai), Thimble (protecting an ally so they can go forward into the Reaches), Kattigan (finds the sorcerer).
Still level 4 as of the end of 4x29: Bolaire, Teor, Julien
*initiated by Aabria - the rest of the Convergence level-ups have been Brennan declaring them level 5, but as Aabria did not level up Thaisha during the Seekers arc she was able to level up here. This is not a bad thing and honestly it fits the pattern for this episode (see below), just an FYI.
Episode 4x27's level-ups are both item-related but ultimately more about realizing a particular interest or calling that also ties back into their nature. Murray is from a gem merchant family but her interest and goals have always been in the realm of the arcane, and these gems, which allow magic at a distance, combine the two. Tyranny is a demon and her purpose is to cause harm, but the knife is a means of channeling that productively, to cause harm to those who have harmed others.
Episode 4x28's level-ups are about facing a difficult but necessary conversation, and at least attempting to make a connection. Thaisha's is regarding Thjazi and also I would argue nature/the world itself with Calling the Corners. Wick's is with his grandmother, Azune's with House Einfasen, and Vaelus's with Hannan. I think all of these also represent a certain degree of honesty with one's self, even if Azune and Wick are very much lying to others.
Episode 4x29 is a complicated one - you can draw a very clear line between Occtis, Kattigan, and Hal in terms of realizing a long-term goal (comprehension of his family's magic; finding Marienna's killer; dedicating the former Dithyramb of Azgra as a theater) but Thimble doesn't quite fit in that. Now I will admit on a meta level that Team Birdwatch - which was entirely level 4 at the start of this episode - is going to be combat-heavy, so your best bet is to make something of a combat triumph rather than waiting for an ideal RP moment. But I also think these are all borne of risking one's self for someone else. Occtis levels up when he touches the veil to protect Thimble from necromantic magic. Kattigan finds Primus because he ran into the aspirants' trap to help Teor. Hal puts on the play despite knowing it might bring down the Candescent Creed on his head because he believes it is what's right for the city and especially for the Rungjani and for Thjazi's wishes. And Thimble risks her life for Julien, also in the service of Thjazi's goals.