The Preposition "Between" in a long sentence

Solution 1:

The sentence is correct, even though I (and I suspect, many others) had to read it twice to be sure I had understood it correctly. To avoid this kind of situation, a writer has to think about how a reader unfamiliar with the background will receive it. Any reader could very well have trouble with nested references. Here we have a massive tripartite subject noun phrase;-

chains of command (between the federal, state and local agencies {involved in law enforcement and CPS})...

The verb phrase that follows explains that these relationship are not always clear. The sentence as a whole is asking the reader to grasp a major feature of the constitutional arrangements in the USA and at the same time understand a problem that arises out of the nature of those arrangements.

If the writer is not subject to fierce character limits, it is easy (and considerate) to find another way.

Because federal, state and local agencies for law enforcement and CPS operate largely independently, chains of command between them are too unclear to deal decisively with unusual, fast-moving crises like the Waco standoff. (235 compared with 175)

For the extra 60 characters you get a sentence that takes you logically through the argument. "Because" (8 characters, including a space) tells you that something is going to be explained by what immediately follows, namely, the fact that the three tiers of agency are largely independent. Now the main clause is free to make the point, referring to the complex subject with the demonstrative pronoun, them.