Oswald may or may not have traveled to Mexico City, but the reasons given make no sense as they are not logical nor rational or even reasonable. Supposedly he went there to get visa that would allow travel to the Soviet Union by way of Cuba. I have problems accepting this because of the following:
1) It is not logical. In January 1961 Oswald, while in Minsk, wrote in his diary: "I am starting to reconsider my desire about staying. The work is drab, the money I get has nowhere to be spent. No nightclubs or bowling alleys, no places of recreation except the trade union dances. I have had enough." Oswald started writing letters to the US embassy in Moscow almost begging to be allowed to return to the United States. He showed some concern about being prosecuted for his defection and had to secure a loan from the State Department to pay for the journey back. All this took 18 months - Oswald, now married and with a young daughter, returned to the USA in June, 1962.
It seems incredible to me that after going to great lengths to get back home, he would change his mind a little over a year later and try to return to the Soviet Union. That isn't rational.
2) In September 1963, Oswald was unemployed and had been for a while. He was so impoverished that he handed over the care for his pregnant wife and child to Ruth Paine - a sinister intelligence operative some believe, but in my view just a housewife with young children of her own. Anyway, he cashed a $30 unemployment check and left for Mexico City, where he applied for a visa at the Cuban Consulate. But how did he plan to get there with just the $30? Had he managed to save some money? Did he expect the Cuban government would make the same arrangements as the Soviets had in 1959-1962? Oswald's actions are again irrational and contrary to logical, reasonable actions.
3) During interrogation on November 22, 1963, Oswald denied having visited Mexico City and I can find no plausible reason why he would lie about it, as it had nothing to do with the crimes of which he was accused.
4) Both Cuban Consul Eusebio Azcue and his Mexican secretary Mrs. Duran denied that the individual who applied for the visa was Oswald. Azcue told the HSCA that 'Oswald' was blond and gaunt, and Silvia Duran told author Anthony Summers, after viewing motion pictures of the real Oswald: "The man on the film is not like the man I saw here in Mexico City. . . . The man on this film speaks strongly and carries himself with confidence. The man who came to my office in Mexico City was small and weak and spoke in a trembling voice. " Like Consul Azcue, she described the visitor bearing Oswald's credentials as 'blond' and 'gaunt' and "short. . . about my size." Silvia Duran was about five feet, three inches tall—at least six inches shorter than the real Oswald.
To be fair and objective, other witnesses claimed that the person they'd observed was indeed Lee Oswald - these include the Soviet officials who dealt with him.
5) The Cuban Consulate and the Soviet Embassy were under 24-hour photo surveillance by the CIA's Mexico City station, and the phones were tapped. After the assassination, a tape recording of Oswald's phone call to the Soviet Embassy and a photo of him were provided to FBI agents in Dallas. But FBI director Hoover told new President Johnson:
LBJ: Have you established any more about the visit to the Soviet embassy in Mexico in September?
Hoover: No, that’s one angle that’s very confusing, for this reason—we have up here the tape and the photograph of the man who was at the Soviet embassy, using Oswald’s name. That picture and the tape do not correspond to this man’s voice, nor to his appearance. In other words, it appears that there is a second person who was at the Soviet embassy down there.
6) An argument in support of the Warren Report: if Oswald didn't visit Mexico City, where the heck did he go? No one saw him in Dallas or New Orleans, as far as I know, except Sylvia Odio - I am not sure about the date though.