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Dabei frequently appears along with entsprechen in DWDS:
Die Referenzversion (ISO 646-IRV) entspricht dabei bis auf eine Position dem ASCII.
Dabei entspricht die Änderung ihren eigenen Prinzipien.
What is the "dabei" doing here? It does not appear in Duden's or DWDS's entries for entsprechen. Is it necessary?
Dabei makes a similar appearance with greifen:
Mehr als 50 deutsche Hooligans griffen dabei gegen 17.30 Uhr in der Nähe des Bahnhofs ukrainische Fans an.
"Dabei" isn't necessarily combined with "entsprechen", but it's just a regular adverb. In the two sentences you quoted, it has (probably) different meanings.
In a more literal sense, "dabei" just means something like "with this", "with regard to this", "in this instance" or something similar. You can think of it as similar to "bei dem", "bei der" etc.
Für das Rezept braucht man drei Packungen des Gewürz'. Dabei entspricht eine Packung fünf Gramm.
For the recipe you need three packs of the spice. (With this / in this instance / in this context), one pack equals five grams.
or, in more natural English
For the recipe you need three packs of the spice. Here, one pack equals five grams.
To give an example with a verb other than "entsprechen"
Alle Reisenden waren aus dem Zug ausgestiegen, aber sie war nicht dabei.
All travellers had exited the train, but she was not (with this / with them).
or, in more natural English
All travellers had exited the train, but she was not among them.
This is probably how "dabei" is used in the first of your two examples
Die Referenzversion (...) entspricht dabei bis auf eine Position dem ASCII.
The reference version (...) corresponds, except for one position, to the ASCII in regard to this.
Without additional context, we can't say what exactly it refers to.
In a more figurative sense, "dabei" can mean something like "obwohl" ("even though", "although")
Die Gläser sind zerbrochen, dabei waren sie sorgfältig verpackt.
The glasses did break, even though they were packaged carefully.
It's hard to say without additional context here, too, but I'd reckon this to the meaning "dabei" is used in the second example
Sie hat gegen die Änderung gestimmt. Dabei entspricht die Änderung ihren eigenen Prinzipien.
She voted against the change. Even though the change adheres to her own principles.
EDIT: Regarding the added third example, here the main verb isn't "greifen" ("to grab"), but "angreifen" ("to attack"). The usage of "dabei" is (as far as we can guess, again for the lack of context) similar to the first example
Gestern gab es Ausschreitungen vor dem Fußballspiel. Mehr als 50 deutsche Hooligans griffen dabei gegen 17.30 Uhr in der Nähe des Bahnhofs ukrainische Fans an.
Yesterday, there were riots before the soccer game. (With those / during those riots), more than 50 German hooligans attacked Ukrainian fans around half past five p. m. near the train station.
