Preconditions for averaging time series I have several time series of annual data from which I'd like to calculate a yearly mean (arithmetic or robust).
Am I correct to assume that each of these time series has to fulfill the conditions of having the same mean and being homoskedastic (i.e. stationarity)? 
Is it also important to prewhiten the individual series (i.e. remove autocorrelation)?
Is there a canonical reference on this topic?
 A: Averaging over time
If you have $m$ series with constant population means $-\infty<\mu_1,\dots,\mu_m<+\infty$*, you can define the population mean of all series as a vector of individual means $\mu:=(\mu_1,\dots,\mu_m)'$.
Then the sample mean over time, $\bar x:=(\bar x_1,\dots,\bar x_m)'$, where $\bar x_i=\frac{1}{T}(x_{i,1}+\dots+x_{i,T})$ is the sample mean of the $i$th series, will be a consistent estimator of $\mu$ if the individual series are stationary. For example, conditional heteroskedasticity and/or autocorrelation of the individual series would not prevent consistency but would reduce efficiency of the estimator $\bar x$.
Averaging across series and over time
In the setting above, you can define the population mean of the average (across "space") of all series as $\mu:=\frac{1}{m}(\mu_1+\dots+\mu_m)$. (You could also extend this trivially to a weighted average.)
Then the sample mean across series of the sample means over time, $\tilde x:=\frac{1}{m}(\bar x_1+\dots+\bar x_m)$, will be a consistent estimator of $\mu$ under stationarity. Once again, conditional heteroskedasticity and/or autocorrelation of the individual series would not prevent consistency but would reduce efficiency of the estimator $\tilde x$.

Other points

Am I correct to assume that each of these time series has to fulfill the conditions of having the same mean and being homoskedastic (i.e. stationarity)?

If you are averaging over time, then you need the (unconditional) mean to be the same over time for each individual series. Without that the estimators would not have meaningful counterparts in population and thus would be estimating some weird objects that you might not be interested in. Homoskedasticity is not required for that, however, as it only affects efficiency but not consistency and not the "meaningfulness". Note also that conditional homoskedasticity is not required for stationarity.

Is it also important to prewhiten the individual series (i.e. remove autocorrelation)?

Not really, because autocorrelation does not prevent consistency; it only results in a loss of efficiency. I am not aware of a way around it as autocorrelation effectively means the sample contains less information about the population mean (which you are trying to estimate) than an i.i.d. sample would. A transformation that could help gain some efficiency would be to account for conditional heteroskedasticity, if any.

Is there a canonical reference on this topic?

Not that I am aware of.

 *This excludes the possibility that one or more of the time series are random walks, because random walks do not have population means. However, under cointegration you could still be fine if the simple unweighted average happens to be a cointegrating combination of the series. 
A: The property of "consistency" of an estimator presupposes the existence of a common theoretical moment that is being estimated through a sample mean.
In general,


Sample means are consistent estimators of the corresponding
    theoretical moments, if the underlying population is   

  
  
*
  
*stationary up to the corresponding moment (1st-order, second-order, etc) 
  
*ergodic
  
*and it is so  along the index over which we average.
  

Note 1: the first condition is essentially a re-statement of the fact that we need the random variables to have common the moment we want to estimate.
Note 2: in many cases, stationarity and ergodicity "go together" (need the same set of conditions), but not always. Ergodicity is a deep mathematical concept, and I do not know how to present it summarily.
Note 3: Condition 3 relativizes the previous two conditions.
Case in point : it appears that in the OP's case, we want to obtain an estimate of the "ensemble mean" (in time series terminology), or in panel-data terminology, to estimate the "cross-sectional" mean in each time period, and form a time series of these cross-sectional means. Looking at the conditions, we require 
$$E(X_{1t}) = ...= E(X_{Kt}) \;\;\; \forall \;t$$
but we can allow 
$$E(X_{1t}) = ...= E(X_{Kt}) \neq E(X_{1,t+j}) = ...= E(X_{K,t+j}),\;\;\; j\in \mathbf Z$$
If the second situation happens, we will obtain a non-stationary time series of averages, but that's life, and it does not affect the properties of each cross-sectional sample mean.
Moreover, do we need the variances to be constant across time? NO, we do not "care" what happens across time, since what we want is to estimate the cross-sectional mean.  
Perhaps we need the variance to be common across "space"? Again, NO, since we are only estimating the first moment, the mean, and we do not "care" about its distribution. Essentially what we need is the conditions for a Law of Large Numbers(LLN) to hold, along the index we are averaging over. If the cross-sections are independent, the only thing we need is that they have a common mean (this is Khinchin's LLN). Regarding the characteristic of independence, it can be relaxed up to a point, and we can start talking about martingales or mixing processes (here along the "space", cross sectional index), and again we will obtain an LLN.
Finally, there may be cases where, given the research purpose/goal it is meaningful to average over random variables that have different theoretical means.
For example assume that we have the Gross Domestic Product (GDP) of different countries. For our purposes we may want to calculate the "average GDP value per time-period", even though we know that most probably for each country the theoretical mean value of its DGP per time period is different (i.e. we do not argue that the DGP of China, viewed as a random variable in each time period, has the same distribution as the DGP of Panama, and that the two represent different realizations from the same distribution). Here, we have Chebyshev's LLN for independent rv's (that the sample average will converge to the average of the true expected values), and also Markov's LLN, which allows for a degree of stochastic dependence and still leads to convergence of the (here) cross-sectional sample average to the average of the true means.
