These issues are really easy to dissect once you stop thinking about them in terms of "Rights infringement" and start thinking in terms of a "Godfather" exacting his long awaited vengeance.
Depending on their specific contract language, both Colbert and Kimmel would be considered "employees" or "independent contractors". Either way, they are not the owners of the shows that they appear on.
Both of their contracts likely have clauses that specify what is and isn't acceptable behavior and what the ramifications are if those clauses are violated. In addition, those contracts will also have "management rights" clauses specifying conditions regarding contract cancelations or contract extensions etc.
Both Kimmel and Colbert have the right to "free speech" like everyone else. That said, those shows do not provide a personal platform for them to say whatever they want about anything and/or everybody without accountability. Additionally, both CBS and ABC have the right to manage their shows within the framework of the contracts that they have signed with Colbert and Kimmel. Very simple.
Now to the real issue:
Trump is as "thin skinned" as they come. His fragile ego and lack of self esteem causes him to attack anyone and everyone that says anything bad or unflattering about him.
Talk show hosts like Colbert, Kimmel, Fallon, Daily etcetera have been on Trump's hit list for years. Now, with the help of a couple of capitulation networks - aided by an Administration full of "hit men" - Trump has taken government weaponization to a never before seen level. So, how did Trump attack Colbert and Kimmel? This one is also easy to figure out.
Skydance (CBS ), owned by Trump's buddy Larry Ellison, needed FCC approval for their billion dollar merger. Since Trump controls the FCC Chairman and hates Colbert, this was simply a matter of leverage or more specifically extortion. Trump promises FCC approval if Colbert is cancelled. Colbert gets cancelled and Skydance gets FFC merger approval. Presto!
Nexstar (ABC) also needs FCC approval for their merger. Trump controls the FCC and also hates Kimmel. Now, another "cancellation" of a Trump critics show is going to look rather suspicious. As such, Trump needs a "red herring" and Charlie Kirk's death provides just that. Kimmel comments on the event, the FCC attacks and presto, Kimmel's show is "suspended" by ABC.
As previously stated, when you think of these two events in crime family terms, neither are "free speech" issues. Absolute free speech doesn't exist. What does exist is Trump's strong desire for vengeance against those who refuse to "kiss the ring", and his current ability to weaponize the government to achieve it.