There is a considerable difference of opinion between Republicans and Democrats about mail in balloting in the United States. Let's look deeper at the issue.
First of all, rules about voting in general are set up by each individual state, with some oversight by the Federal Government. The oversight is pretty small, mostly to the level of ensuring the ballot is fair. They do specify the date, the first Tuesday after the first Monday. It is set as such to not be on the first day of November. Tuesday has been election day since the beginning of the country, primarily to allow for travelling to the election site on Monday and be back by Wednesday to continue working on their farms. The rules are very loose, just a framework really. They primarily allow one to provide aid if disabled, ensure that everone of legal age can vote, and ensure that blatent discrimination does not happen.
The vast majority of states allow absentee voting, which is voting whereby one can vote before election day. This can be done either in person at specifically designated areas, or via mail in balloting. 10 states automatically will mail to all eligible voters, including mostly states that tend to vote Democractic, but at least a few battleground and even some Republican leaning. Many states require an excuse to request a mail in ballot, although this year COVID-19 fears is considered a valid reason. Some states allow one to request a ballot online, others require one to physically mail in a request. In every instance, a voter must be registered prior to requesting a mail in ballot.
The concern expressed primarily by Republicans is over voter fraud. To be certain, voter fraud can and does happen, although the amount of it is highly questionable. According to the Brennan Center, the actual amount of voter fraud sits at 30 votes/ million votes cast. The closest Presidential election in recent history was the election of 2000, where George Bush won with a 537 vote count with almost 6 million votes. At that rate of fraud, even that super close election did not have its count altered.
The easiest way that one could commit voter fraud would be to vote in person and mail in a ballot, but every state checks for that, and in fact most of the fraudlent votes are something along those lines. There is more of a question if the votes should happen in different states, but that is a very difficult problem to resolve. In my opinion, if one moves one's state of driver's license they shoudl automatically revoke the old state's voter registration as well, but this does not happen. In fact, once I had moved several years prior, received a jury summons to appear across the country, informed them that I had moved, and then the voter registration office from that state requested that I mail in a form to cancel my voter registration in that state! For that state, Arizona, one can do many election related things online, it was strange that this couldn't be. In any case, I just did nothing, and presumably am no longer registered. This makes sense because in many states one is registered to vote at the same time one receives a driver's license. In fact, as it is federal law to offer voter registration to eligable voters when receiving such an ID card it seems it could be tracked.
Another point of some interest. If a voter does not actually vote for a number of years they are placed on an inactive voter list. That list requries additional ID to vote. The ID must contain an address, a driver's license would be fine, proving they still live in the precint.
The other claim that often comes from Republicans is that of illegal immigrants voting. It turns out that pretty much every state does not allow for those who are not US citizens to vote. One does not, however, have to provide a birth certificate, state identification (Driver's license, etc), or other similar form of ID to prove they are eligable. Still, this seems to be limited, although it does happen. According to this paper, although if caught they could be charged with a crime. And there is enough evidence to do so, given enough time. Most states require some form of photo ID, which acts to counteract that. In North Carolina, a state which does not require showing ID to vote, a total of 19 cases of foreigners voting were brough to court, of 4.7 million votes cast. Many do end up inadvertently registered to vote, when they receive a Driver's ID card, but most correctly do not actually vote, as that would be illegal.
I will save it for another day on why such talks are so persistant, and why the Democrats in general push for more voting. But from all reports, while voter fraud can and does happen, the actual number of cases is in the dozens per million votes cast, and is not really significant.