This is not an easy exercise but I will try to explain it as simply as possible. The prohibitions against bills of attainder and ex post facto laws are found in Article I, Section IX paragraph 3, with respect to the federal government, and Section X, Paragraph 1, with respect to state governments.
Bills of Attainder are extremely rare and there is little to no case law on them. A bill of attainder is a law that is an act by the legislature that declares a person or group of people guilty of a crime and imposing a punishment without a trial. For instance, if the legislature passed a law declaring all persons who voted for Trump guilty of treason and sedition and incarcerates or executes them. There is absolutely no due process in the execution and enforcement of the law.
An ex post facto law is a law that makes some act done in the past illegal. If congress passed a bill today making it illegal to purchase a firearm during a the current state of emergency and made the law retroactive to the declaration of the state of emergency making anyone who purchased a firearm since mid March a criminal, even though it was legal at the time. Ex post facto laws would still require a trial (due process) if they were legal.
I do not know when federal background checks began across the country, but I know I had a background check and a 2 week waiting period in Pennsylvania in 1984 when I purchased my first firearm in Pennsylvania. As best I can tell, prior to the Brady Bill, made law in 1993, effective February 28, 1994, background checks were done at a state level. To be honest with you, I know I had one but do not remember if I had to get finger printed or what was done.
The Brady bill made it a federal requirement to have a background check to purchase certain firearms. Anyone who purchased one prior to that point was not suddenly guilty of a crime for bypassing a process that did not then exist. That does not mean they owned the firearm legally. If there was a law that prohibited them from possessing a firearm when they first got it, it was illegal then, continued to be illegal. This is neither an ex post facto law nor a bill of attainder.
The question everyone want to really know the answer to is if they were to pass a law that required everyone to turn in their so called “assault weapons” and “high capacity magazines,” by a certain date, would that violate the Constitution? It really becomes a question of can the legislature pass a law that makes having something that was legal yesterday illegal tomorrow?
Such a law would not be a bill of attainder or an ex post facto law. It is not making you a criminal for doing or having something in the past. It is making it illegal to possess something at some point in the future. It sounds bad and would not be a good idea for government to do that, but it is possible. The coca plant was initially legal in the US and was used in many products, most notably Coca Cola! In 1914, cocaine became illegal in the US. What was legal in 1913, was illegal in 1914.