About the simplest thing you can do is interpolate normalized counts over time and (almost) the simplest form of interpolation is linear.
Specifically, suppose $y_i$ is the state population at time $i$ and $x_i$ is some other count (by age, tract, or whatever). Define $\xi_i = x_i/y_i$. Suppose $i$ is a year for which you do not have the periodic data. Let $i_{-}$ and $i_{+}$ be the years immediately preceding and following $i$, respectively, for which $x_i$ is available. The linearly interpolated estimate of $\xi_i$ is
$$\hat{\xi}_i = \frac{\xi_{i_{-}} (i_{+} - i) + \xi_{i_{+}} (i - i_{-})} {i_{+} - i_{-}} \text{.}$$
The estimate of $x_i$ is
$$\hat{x}_i = \hat{\xi}_i y_i.$$
The sums will come out correctly because this estimator is linear with weights summing to unity. For example, suppose you are tracking two variables $x$ and $z$ which count complementary parts of the population (such as males and females), so that $x_i+z_i = y_i$ whenever you have all three counts. Defining $\xi_i = x_i/y_i$ as before and, similarly, $\zeta_i = z_i/y_i$, the two fractions sum to unity: $\xi_i + \zeta_i = y_i/y_i = 1$ for all $i$. Therefore the interpolated fractions also sum to unity:
$$\hat{\xi}_i + \hat{\zeta}_i = \frac{\xi_{i_{-}} (i_{+} - i) + \xi_{i_{+}} (i - i_{-})} {i_{+} - i_{-}} + \frac{\zeta_{i_{-}} (i_{+} - i) + \zeta_{i_{+}} (i - i_{-})} {i_{+} - i_{-}}$$
$$= \frac{(\xi_{i_{-}} + \zeta_{i_{-}}) (i_{+} - i) + (\xi_{i_{+}} + \zeta_{i_{+}}) (i - i_{-})} {i_{+} - i_{-}}$$
$$= \frac{(i_{+} - i) + (i - i_{-})} {i_{+} - i_{-}}$$
$$= 1.$$
Whence $\hat{x}_i + \hat{z}_i = y_i(\hat{\xi}_i + \hat{\zeta}_i) = y_i$ as desired. This generalizes to population partitions of any size, such as age distributions.
