Does it logically follow that gases compressed to higher pressures move faster?

It should. They are gases, with kinetic energy, are they not? So if they should get compressed to higher pressures, they thus move faster.

It's straightforward to show this for ideal gases; consider the RMS speed:

#v_(RMS) = sqrt((3RT)/M)#

where #R# and #T# are known from the ideal gas law (#R# in #"J/mol"cdot"K"#) and #M# is the molar mass in #"kg/mol"#.

Well, as I hinted, you can use the ideal gas law to rewrite this.

#PV = nRT#

#=> RT = (PV)/n#

Therefore:

#color(blue)(v_(RMS)) = sqrt((3PV)/(nM))#

#= color(blue)(sqrt((3PV)/(m)))#

where #m# is the gas mass in #"kg"#.

From this, we can see the pressure-dependence of the RMS speed for ideal gases.

With some rearrangement:

#P = (mv_(RMS)^2)/(3V)#

#= 2/3 K_(avg)/V#

And so:

#color(blue)(K_(avg) = 3/2 PV)#

And here we can also see the pressure-dependence of the average kinetic energy of translation. To show that this is correct, we know that for ideal gases we get:

#color(blue)(K_(avg) = 3/2 nRT)#

which is the equipartition theorem for ideal gases in translational (linear) motion. It also shows why we say that temperature is proportional to the average kinetic energy.