The answer to the question (a counterexample)
Properties of random processes will in general be time-dependent. They are not only when talking about stationary processes. Another related concept (not relevant here but that you might find interesting) is ergodicity. Most of us find it difficult to understand the difference between both concepts, but then you find this brilliant Dilip's answer to clear out all doubts.
Going back to your question: take for example the simple random process
$$X(t)=At$$
where $A$ is some random variable with mean $\mu_a$. The mean of the process would then be:
$$\mathbb{E}[X(t)]=\mathbb{E}[At]=t\mathbb{E}[A]=t\mu_a$$
It's easy to see that the mean depends on $t$, thus it's time-dependent. I hope this easy example shows why, in general, properties of random processes vary with time.
Additional info that might help
I think your confusion arises from thinking that to build a stochastic process you grab many realizations of a single random variable and put them in a sequence to represent a time-signal. That's not how a random process is built in general. If you do this, then the mean will not vary with time because the process will be stationary. But if you don't, it may or may not vary.
I'll use an example here to make myself clear. Suppose you have a random variable $Z$ that represents the result of throwing a fair 6-faced dice. Then $\mu_z = 3.5$. If you throw the dice 3 times and you put them together to form a discrete-time signal of 3 samples, then you will have the following random process:
$$X(n)=Z_1\delta(n) + Z_2\delta(n-1) + Z_3\delta(n-2)$$
where $Z_i$ represents the random variable corresponding to the $i$-th throw of the dice.
Note that each sample of $X(n)$ is a random variable. This is correct for any random process in general. For a given random process $X(n)$, $X(n_0)$ for any $n_0$ will be a random variable.
On the other hand, once you've thrown the dice 3 times, you are not dealing with anything random anymore: you have a signal. That is, a realization of the random process.
If you build a random process such as we have just built $X(n)$, then you are right: the mean of the process will be constant and equal to the mean of the random variable used for each sample (as in this case). In fact, all properties will be constant and equal to those of the variable. But that's just in that specific case, because all samples of the process will be i.i.d., and thus, as @Fat32 has explained in the comments, the random process will be independent of time ($t$). In our dice example:
$$\begin{align}
\mathbb{E}[X(n)]&=\mathbb{E}[Z_1\delta(n) + Z_2\delta(n-1) + Z_3\delta(n-2)]\\
&=\mathbb{E}[Z_1]\delta(n) + \mathbb{E}[Z_2]\delta(n-1) + \mathbb{E}[Z_3]\delta(n-2)] \\
& = \mu_z \qquad \text{for any} \ n\in\{0,1,2\}
\end{align}$$
But this will not always be the case with random processes, such as in the example I wrote at the top of the answer, in which every random variable obtained by evaluating the random process at different values of $t$ is different from all others, therefore preventing the process from being i.i.d. and making its statistical properties time-dependent.
