Most research e.g. summarized by Cornelis et al. (2009) suggests that most people become more conservative as they age:
The finding that older people tend to be more conservative emerged
from many studies (e.g., Feather, 1977; Grant, Ross, Button, Hannah,
& Hoskins, 2001; Henningham, 1996; Maltby, 1997; Ray, 1985;
Truett, 1993; Wilson, 1973).
The case of US justices highlighted in another answer thus might not be representative of the population at large. But you also have to keep in mind that most such studies are cross-sectional, which simply means that older people are more conservative than than younger ones at a given point in time. Longitudinal studies are fewer in number.
Cross-sectional survey data have shown moderate to substantial
correlations between chronological age and measures of cultural
conservatism (see Table 1).
Grant and colleagues (2001) found evidence
for both a linear and quadratic effect of age on conservatism,
with scores initially increasing slowly, and then at an accelerated
pace in older age, a pattern also observed by Truett (1993). Previous
studies generally failed to report significant relationships between
age and indicators of economic conservatism (e.g., Duriez & Van
Hiel, 2002; Felling & Peters, 1984; Middendorp, 1991; van Berkel-
Van Schaik & van Snippenburg, 1991; Van Dam, 1993).
The study of Cornelis et al. (conducted in Poland and Belgium) also found support for this linear relationship of conservatism with age, but a very limited quadratic effect. Their explanation is that Openness to Experience (a Big Five personality factor) decreases with age. Additionally they found a more obscure personality factor, namely Need for Closure to be another explanatory link.
Need for Closure (Kruglanski & Webster, 1996; Webster & Kruglanski,
1994) refers to the desire or need for any firm belief or answer,
as opposed to further sustaining ambiguity.
Likewise a 2013 Europe-wide study by Robinson found similarly:
The value categories measured by the ESS [European Social Survey] are the following: conservative values (tradition, conformity and security), openness to change values (self-direction, hedonism and stimulation), self-transcendent values (universalism, benevolence) and self-enhancement values (power, achievement). Of the ten lower order values, tradition shows the strongest positive relation with adult age, while the value of stimulation shows the strongest negative relation with age. With regards to the four higher order value categories, conservative values increased across age groups, while openness to change values decreased. Neither of these value types showed cohort or gender differences. [...] Age effects on the four higher order value types were replicated across all 12 countries in the sample