Since you did not supply any data, we'll start by making some up -- here's a function x[t] that gets noisier over time, and then it is sampled to give the data xsam:
x[t_] := Sin[t] + RandomReal[{-t/2, t/2}];
xsam = Table[x[t], {t, 0, 10, 0.01}];
A moving average can be implemented as the product of n samples of the data and a constant array of length n. Your goal is to increase n as time progresses. One way of implementing that is to have n increase linearly:
xfilt = Table[n = Ceiling[i/10];
Total[xsam[[i ;; i+n]]*ConstantArray[1/(n+1), n+1]],
{i, 1, Length[xsam]-100}];
Here is a plot of the samples and the filtered version:
ListPlot[{Flatten[{ConstantArray[0, n], xfilt}], xsam}]

You can choose any rate of change for the filter length n by changing the functional relationship between n and i.
You asked for a moving average filter to do the smoothing. You might also consider an autoregressive filter for this task. Here is an analogous structure that uses alpha as a smoothing parameter -- I've started with 0.8 (little smoothing) and increased it to 0.99 (lots of smoothing). You can change that by redefining the alpha sequence.
xfilt = ConstantArray[0, Length[xsam]];
alpha = Range[0.8, 0.99, (0.99 - 0.8)/(Length[xfilt] - 1)];
Table[xfilt[[i + 1]] =
alpha[[i]]*xfilt[[i]] + (1-alpha[[i]])*xsam[[i]], {i, 1, Length[xsam]-1}];
ListPlot[{xsam, xfilt}]

One advantage of this approach is that you no longer have a varying time delay in the operation of the filter.