Which time series analysis is appropriate for pooled time series data analysis?
It really depends on the outcome you want from your analysis.

The most straightforward approaches might be resampling:


Interpolate the 10-year census data to month granularity. The interpolation might be linear, polynomial, etc. This effectively imposes a strong assumption in the dynamics of the data. This may make sense for some variables but not for others. And you are probably introducing a lot of variance in the obtained conclusions.
Aggregate the monthly time series into 10-year granularity. This may not be very useful if you want to predict something with an horizon less than 10 years...