[*Spoilers various comics including: Superman Reborn (2017) DC Versus Marvel (1996), and Superman vs. The Amazing Spider-Man (1976), among others.]
Inter-company comic book crossovers are when two companies combine characters from their respective brands. Yet, such crossovers are often set outside of the ongoing continuity. For example Superman vs. The Amazing Spider-Man takes place in a shared universe where they just hadn’t met. However, shared universes are just one way these crossovers can take place. Another way is when characters crossover from their universe into another universe (DC versus Marvel).
Marvel Team-Up #74 (1978). Cover art by Dave Cockrum and Marie Severin. Cover lettering by Irving Watanabe.
It is this second method that can affect an ongoing continuity. One example of this is DC Universe Vs. Masters of the Universe (2014) which mixes DC’s characters with Mattel’s. Yes, DC’s “New 52” continuity is no longer in effect thanks to things like Superman: Reborn, but it firmly interrupted both brands’ comic continuities. This crossover also brings us to the conundrum of whether changes in one continuity (such as with the initiative DC Rebirth [2016-2017]) effectively negates the story’s events for both brands. If it does, then those who read and keep track of these fictional histories (such as Michael Kooiman, the webmaster of Cosmic Teams) might feel cheated of time and/or money.
There is also the problem that publishers can’t fully address events found in the comics after the crossover. Such is the case with how writer Kurt Busiek could only somewhat reference the events from JLA/AVENGERS (2004) in JLA Vol. 17: Syndicate Rules (2005). This kind of problem occurs due to various preliminary decisions such a how profits will be divided up, publishing rights (such as copyrights and trademarks), and deciding on which company is officially publishing the comic. For these reasons, crossovers usually are kept insulated so that they do not disrupt continuity.
That all said, there was a brief success with this type of crossover working out. This occurred during the last few years of Archie Comics having the licensing rights to publish Sega’s Sonic The Hedgehog (Sonic The Hedgehog [1992-2016). During this time Archie Comics acquired the rights to Capcom character Mega Man (Mega Man [2011-2015]. Archie Comics then published two crossovers starring the characters with both series’ ongoing continuity being affected (Sonic/Mega Man: Worlds Collide: The Complete Epic  and Sonic/Mega Man: Worlds Unite 1: Deadly Fusion ). Though when both series got canceled, it caused this success to fizzle out a bit. In conclusion, crossovers are most successful when they exist as standalone tales.