First of all, this entire discussion assumes that Alabama's law will be upheld, and Roe will be overturned. Roe is still good law, and under it the Alabama law is pretty clearly invalid. However it is possible that the US Supreme Court will overrule itself and Roe will no longer apply. The Court has overruled itself in the past, perhaps most rapidly and thoroughly in West Virginia State Board of Education v. Barnette 319 U.S. 624 (1943), (the second flag salute case) which overturned Minersville School District v. Gobitis (1940). Gobitis involved a very similar law and almost exactly similar facts, and there had been only one change in the membership of the Court. Moreover the West Virginia law included extensive quotes from Gobitis and was clearly intended to follow that decision closely. But the Court changed its ruling totally in only 3 years (from 8-1 to 6-3 the other way).
As discussed at length in this Wikipedia article Freedom of Movement is a right protected under the US constitution. It has been recognized as such at least since Paul v. Virginia, 75 U.S. 168 (1869), and is traced to a right guaranteed under Article Four to the Articles of Confederation prior to the current US Federal Constitution. Travel between states is now protected by the Privileges and Immunities Clause. See Saenz v. Roe, 526 U.S. 489 (1999). See also This article on "The Right to Travel" from the Legal Information Institute.
Therefore, a state law attempting to prohibit a person from traveling out of state (or out of the US) to obtain an abortion would be unconstitutional. To the best of my knowledge, neither the Alabama law nor any other recently passed or seriously proposed law purports to impose such a prohibition.
A federal law to this end would surely be contested, and might well be held unconstitutional, but there is no case in direct point that I know of. The federal Mann act did once prohibit interstate "transport of females" for "immoral purposes" (generally sex) and now criminalizes interstate travel for purposes of illegal prostitution. A similar law to prevent travel to obtain an abortion might be upheld, no one can be sure until one is passed and challenged.
If a person traveled outside a state, such as Alabama, that prohibited abortion, to one that permitted it, and had an abortion there, and then returned to the original state, an attempt to prosecute such a person would be highly likely to fail. First of all, the current Alabama law does not purport to prohibit out-of-state abortions. Secondly, such a prosecution would probably be held to be a burden on the right to travel. There is also the history of people traveling to other states to obtain divorces not lawful in their own states. Attempts to hold such divorces invalid did not succeed. The hypothetical case seems similar.