Z-score normalization, as you have already guessed, cannot deal well with non-stationary time series since the mean and standard deviation of the time series vary over time.

Min-max and another commonly used normalization in stationary data, the decimal scaling normalization depend on knowing the maximum values of a time series.

The most commonly used method for data normalization of non-stationary time series is the sliding window approach (J. Lin and E. Keogh, 2004, Finding or not finding rules in time series). In short:

The basic idea of this approach is that, instead of considering the
complete time series for normalization, it divides the data
into sliding windows of length ω, extracts statistical
properties from it considering only a fraction of ω
consecutive time series values (H. Li and S. Lee, 2009, Mining frequent itemsets over data streams using efficient window sliding techniques, Expert Syst. Appl., v. 36, n.
2, p. 1466-1477. J.C. Hull, 2005), and normalizes each
window considering only these statistical properties. The
rationale behind this approach is that decisions are usually
based on recent data.
(...)
The sliding window technique has the
advantage of always normalizing data in the desired range.
However, it has a drawback of assuming that the time series
volatility is uniform, which is not true in many phenomena

Another more advanced and less used (so far) is Adaptive Normalization

can be divided into three stages: (i)
transforming the non-stationary time series into a stationary
sequence, which creates a sequence of disjoint sliding
windows (that do not overlap); (ii) outlier removal; (iii) data
normalization itself.

Check the link on Adaptive Normalization and all its references, there is relevant information in there.