"cascaded" cross-correlation
I'm afraid your statement isn't true. This can best be seen in a suitable choice of basis, one that simplifies the cross correlation. This basis if of course the shift invariant periodic Fourier basis on your support interval.

Let's label the basis vectors $F_n$ for integer $n$. The cross correlation of two different such basis vectors vanishes, because the vectors are orthogonal and shift invariant: $\langle F_n,F_m \rangle =0$ for $m\neq n$. If we slightly generalize the cross correlation to work for complex signals as in $$\langle A,B \rangle (\tau):=\int_{-\pi}^\pi A^*(t)B(t+\tau)dt$$ and assume normalization of the Fourier basis, then $\langle F_n,F_n \rangle (\tau)=F_n(\tau)$.

With this, we can construct your $x(t)$,$y(t)$ and $z(t)$ signals like follows:
$$x(t) := c_{xy} F_{n}(t) + c_{xz} F_{k}(t)$$
$$y(t) := c_{yx} F_{n}(t) + c_{yz} F_{m}(t)$$
$$z(t) := c_{zy} F_{m}(t) + c_{zx} F_{k}(t)$$

with mutually different integer indices $k,m,n$. The complex coefficients $c_{ab}$ for $a,b\in\{x,y,z\}$ can be chosen arbitrarily and independently.

The corresponding correlation functions are:

$$\langle x,y \rangle (\tau) = c^*_{xy} c_{yx} F_n(\tau)$$
$$\langle y,z \rangle (\tau) = c^*_{yz} c_{zy} F_m(\tau)$$
$$\langle x,z \rangle (\tau) = c^*_{xz} c_{zx} F_k(\tau)$$

So each cross correlation function is a complex sinusoid with a phase factor in front. The phase factors and the sinusoids of all three correlation functions can be chosen absolutely independently. If you want to talk about maxima instead of phase factors you can take the whole setup to the reals by either adding the conjugate basis functions or just take the real part of these correlation functions. Either way, you can chose the coefficients to give you any set of maximum lags $\tau_{xy,zy,xz}$ without the requirement of the connection between them you conjectured.

So as requested let's get more explicit. The construction principle that guarantees independence of the three cross correlation functions should be relatively obvious: The orthogonality of the Fourier basis allows us to construct the signals so that the signals $x,y,z$ only overlap pairwise. In other words, the example above demonstrates that there is no transitivity for cross correlations that would allow general statements about $\langle x,z \rangle$ from $\langle x,y \rangle$ and $\langle y,z \rangle$.

The simple construction from above makes them overlap with just one basis function, which is a complex sinusoid. This is probably why the result cannot be immediately seen to apply to the original question. If you want maxima we need to use real numbers, and single real basis functions also do not have a single maximum, so we need to have more than one basis function of overlap between the signals (except for $F_1$, which already only has one maximum per period).

The simplest full example that I can come up with right now and that follows the design principle described above is:
$$x(t)=\sin(t)+\cos(4t)+\cos(5t)$$
$$y(t)=\sin(t)+\sin(2t)+\sin(3t)$$
$$z(t)=\sin(2t)+\sin(3t)+\sin(4t)+\sin(5t)$$

The base indices that overlap between the signals are $1$ for $xy$, $2$ and $3$ for $yz$, and finally $4$ and $5$ for $xz$. The latter have a phase shift introduced to realise your requirement. The overlaps group adjacent basis functions so that we have a single guaranteed maximum.

At this point it's probably easier to inspect the resulting correlation functions graphically.

First, $\langle x,y \rangle$:



$\langle y,z \rangle$:



and finally the independent third pair $\langle x,z \rangle$:



The maxima of the first two cross correlations are clearly at $\tau=0$, while the third pair correlates best at some $\tau&gt;0$.

This example works in both discrete and continuous time, as it only uses properties of the Fourier basis that are realised in both domains.