You can only acquire a target lock up to range three. Does that mean that you loose a target lock if the ship it is on moves further away?

Even though you can (normally) only acquire at target lock at range 1-3, you can maintain a target lock at any range.

Welcome to the forums!

As FireSpy said, you are normally restricted to a maximum of R3 when acquiring the TL but once you have it the token stays on the target until you spend it, have the ship TL another target, or the ship is destroyed.

There are upgrades to Lambda and epic ships that would allow TLs beyond R3.

There are also upgrades that will allow a ship to have multiple TLs on targets.