No, to keep the “feel” (playing diatonic), you would continue with the parent scale. So C Lydian to E Aeolian. That’s not to say that you CANT do what you mentioned, but you would lose the Lydian vibe and you would modulate the mode.

I want to clarify here, the way you approach modes on a per chord basis is fine and needed, but you have to shift your perspective to see the side where they are an alteration, this way so you can modulate modes and be more tasteful in how you use them.

If you’re playing C Ionian-F Lydian-G Mixo, for the sake of navigation and visualization of a roadmap on your instrument, that’s fine. But understand that you’re really only playing in one mode when you do that, which is dependent on the chord progression.

They’re literally all the same notes, so you have to ask yourself, if they’re all the same notes, how do I know what mode I’m playing in? You have to look at the chord progression firstly, see what feels like home, what is the tonic?

You can approach modes like that where you stick strictly diatonic (same parent scale), but modes/music is used beyond sticking to one key/mode. The magic of modes is also in how you mix/modulate them, how you add more flavor to music. How modulate from one mode into another.

Typically when modulating between modes you want to play a minimal amount of chords like going back and forth between 2 chords or just a vamp. The less you have going on underneath, the more freedom you will have with your improvising over the chords, so it’s a good place to start.

It’s good to perceive them as alterations so you can really know what modes would be more pleasing when modulating. For example C Dorian to C Aeolian will have a one note difference, so it can sound very tasteful if you weave in and out of it. C Major to C Locrian would be super abrupt and likely not pleasing, at least not for weaving in and out of each other, there are a lot of different notes their. Say there’s a C vamp and you play D Dorian then A aeolian, it’ll still all sound like C major, because the vamp is emphasizing the C, as a chord progression resolves and emphasizes the tonic.