Frankly, I also searched for but couldn’t find much of others addressing the parallels of Ruth and Elisha. Walfish was almost always the writer. What I did find were mostly studies of one book that had cross references to the other(s) (e.g. Ruth noting a Kings book or vice versa). For some other material I found, see the comment at the bottom of this answer.
I was surprised with the quantity of parallels, especially with Ruth having only four chapters, and how specifically the parallels matched. They seem beyond coincidental, both apparently stressing how important it is to be mentor, or to learn from, properly serve, and be loyal to a mentor. In each case the follower (Ruth or Elisha) is well rewarded and becomes able to reward others who treat him/her well.
This list of parallels has some you and Bench have mentioned.
Famine, drought, and poor widow providers: Naomi (Ruth’s mentor) travels with her two daughters-in-law after a famine; realizing and stating she’s destitute, Naomi has to have a poor widow provide for her once they reach Bethlehem. The widow Ruth provides. (Ruth 1:21, 3:17)
When Elijah (Elisha’s mentor) has a drought, the Lord tells him “I have commanded a widow there to provide for you.” (i.e. the widow/mother who has only a handful of flour and a little oil). 1 Kings 17:7-9 (NASB)
Three important men die: Naomi, whose husband had died ten years earlier, has both of her sons die ten years later. She therefore moves away with her daughters-sin-law (Ruth and Orpah), traveling toward Bethlehem. (Ruth 1:3-5)
Elijah has the king send a captain to order him down from the hilltop; because Elijah is “a man of God”, fire comes down from heaven and consumes the captain. That soon happens to a second captain. Then the king also dies. (2 Kings 1: 9-12)
In the case of Naomi and Ruth the three men who died were as close as a father, husbands, and sons. In the case of Elijah and Elisha, they weren’t relatives, but the mentor/”father” Elijah died not long afterward.
“Return” ordered 4 times, “Stay here” 3 times, “Go back” once: When Naomi no longer wants Ruth and Oprah following her, she 3 times tells both daughters-in-law to “Return” to their mothers/families. With Ruth the one who won’t leave her, Naomi tells Ruth to return a fourth time. (Ruth 1:8, 11, 12, 15)
Ruth 1:8-16 (NASB)
And Naomi said to her two daughters-in-law, “Go, return each of you to her mother’s house. May the LORD deal kindly with you as you have dealt with the dead and with me. 9 May the LORD grant that you may find rest, each in the house of her husband.” Then she kissed them, and they lifted up their voices and wept. 10 And they said to her, “No, but we will surely return with you to your people.” 11 But Naomi said, “Return, my daughters. Why should you go with me? Have I yet sons in my womb, that they may be your husbands? 12 Return, my daughters! Go, for I am too old to have a husband. If I said I have hope, if I should even have a husband tonight and also bear sons, 13 would you therefore wait until they were grown? Would you therefore refrain from marrying? No, my daughters; for it is [a]harder for me than for you, for the hand of the LORD has gone forth against me.”
14 And they lifted up their voices and wept again; and Orpah kissed her mother-in-law, but Ruth clung to her.
15 Then she said, “Behold, your sister-in-law has gone back to her people and her gods; return after your sister-in-law.” 16 But Ruth said, “Do not urge me to leave you or turn back from following you; for where you go, I will go, and where you lodge, I will lodge. Your people shall be my people, and your God, my God. 17 Where you die, I will die, and there I will be buried. Thus may the LORD do to me, and worse, if anything but death parts you and me.”
When the mentor Elijah no longer wants Elisha following him, he asks him to “Stay here” three times.
2 Kings 2:4-6 (NASB)
2 Elijah said to Elisha, “Stay here please, for the LORD has sent me as far as Bethel.” But Elisha said, “As the LORD lives and as you yourself live, I will not leave you.” So they went down to Bethel. 3 Then the sons of the prophets who were at Bethel came out to Elisha and said to him, “Do you know that the LORD will take away your master from over [a]you today?” And he said, “Yes, I know; be still.”
4 Elijah said to him, “Elisha, please stay here, for the LORD has sent me to Jericho.” But he said, “As the LORD lives, and as you yourself live, I will not leave you.” So they came to Jericho. 5 The sons of the prophets who were at Jericho approached Elisha and said to him, “Do you know that the LORD will take away your master from over [b]you today?” And he [c]answered, “Yes, I know; be still.” 6 Then Elijah said to him, “Please stay here, for the LORD has sent me to the Jordan.” And he said, “As the LORD lives, and as you yourself live, I will not leave you.” So the two of them went on.
Previously, when Elisha first decided to follow, Elijah told him to “Go back” to plowing. (1 Kings 19:20)
Only the end of life on earth will separate:
When Naomi repeatedly tells Ruth to go back, Ruth firmly replies “Do not urge me to leave you”, but then Ruth even says “Where you die, I will die, and there I will be buried. Thus may the LORD do to me, and worse, if anything but death parts you and me.” (Ruth 1:17) (NASB)
Each of the three times Elijah asks his follower to “stay here”, Elisha strongly replies with “I will not leave you”. Elisha is well aware that he is about to lose his master, and the guild prophets even confront and remind him “Do you know that the LORD will take away your master from over [b]you today?” And he said, “Yes, I know; be still.”(2 Kings 2:3) (NASB)
Leaving behind family and more:
Naomi tells Ruth the four times to go to her family, to be with them and possibly find a husband. She never does. (Ruth 1:1-16)
Elisha kisses and leaves his parents behind, along with his 12 yoke of oxen. He even slaughters the oxen for the good of his people. (1 Kings: 19:20)
Always following and servicing:
Ruth always decides to remain loyal, obey and follow her mentor, Naomi, ready to act as a servant (i.e. except when she’s told to leave). (e.g. Ruth 3:6)
Likewise, Elisha remains loyal to Elijah, always ready to service him (i.e. except when he’s told to leave).
Followers greatly rewarded:
Since Ruth “did just”/exactly what her mentor and the gentleman kinsman wanted, along with helping to provide, she was rewarded with a prominent husband and child son. (Ruth 2:11-14, 3:15)
Elisha also did what his mentor expected. He was rewarded with what he asked for, a “double portion of” Elijah’s spirit. After the chariot took Elijah into heaven, Elisha picked up the mantle and became able to divide and cross the water, heal bad water, bring a dead child back to life, etc. (2 Kings 2:13-14, 19-22, 4:35)
Followers reward prominent people’s hospitality:
The prominent kinsman, Boaz, noticed everything Ruth had been doing, such as the gleaning after the harvesters. He treated her well, as in protecting her; he invited her to eat with him. (Ruth 1:11-14) She later became his wife and gave birth to their son. (Ruth 4:13)
Elisha once had a rich woman show him some hospitality too. She urged him to dine with her, and he did. Later when he passed by he was also allowed to dine with her. (2 Kings 4:8-9) At one point Elisha promised that she’d be holding a son at that time next year. She gave birth to a son, who died a few years later, but Elijah helped bring him back to life. (2 Kings 4: 15-20, 31-36)